This 10 is equal to ten of the units expressed by 1. It is, however, but a single ten, and in this sense may be regarded as a unit, the value of which is ten times greater than the unit expressed by 1. It is called a unit of the second order.

6. When two figures are written by the side of each other, the one on the right is called the place of units, and the one on the left, the place of tens, or units of the second order. Each unit of the second order is equal to ten units of the first order.

When units simply are named, units of the first order are always meant.

Two tens, or two units of the second order, are written
Three tens, or three units of the second order, are written
Four tens, or four units of the second order, are written
Five tens, or five units of the second order, are written
Six tens, or six units of the second order, are written
Seven tens, or seven units of the second order, are written
Eight tens, or eight units of the second order, are written
Nine tens, or nine units of the second order, are written

20

30

40

50

60

70

80

90

These figures are also read, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety.

The intermediate numbers between 10 and 20, between 20 and 30, &c., may be readily expressed by considering the tens and units of which they are composed. For example, the number twelve is made up of one unit of the second order and two of the first. It must therefore be written by setting 1 in the place of tens, and 2 in the place of units; thus,

Eighteen has 1 ten and 8 units, and is written
Twenty-five has 2 tens and 5 units, and is written
Thirty-seven has 3 tens and 7 units, and is written
Fifty-four has 5 tens and 4 units, and is written.

12

18

25

37

54

6. When two figures are written by the side of each other, what is the place on the right called? The place on the left? When units simply are named, what units are meant? How many units of the second order in 20? In 30? In 40? In 50? In 60? In 70? In 80? In 90? Of what is the number 12 made up? Also, 18, 25, 37, 54?

Hence, any number greater than nine, and less than one hundred, may be expressed by two figures. The right hand figure will express units of the first order, and the other, units of the second order.

7. In order to express ten units of the second order, or one hundred, we have to form a new combination.

100

It is done thus, by writing two ciphers on the right of 1. This number is read, one hundred. Now this one hundred expresses 10 units of the second order, or 100 units of the first order. But the one hundred is but an individual hundred, and in this light may be regarded as a unit of the third order. We can now express any number less than sand.

For example, in the number three hundred and seventy-five, there are 5 units, 7 tens, and 3 hundreds. Write, therefore, 5 units of the first order, 7 units of the second order, and 3 of the third; and read from the right, units, tens, hundreds.

In the number eight hundred and ninetynine, there are 9 units of the first order, 9 of the second, and 8 of the third, and it is read, units, tens, hundreds.

one thou

cohuns.

tens.

units.

∞huns.
Cotens.
counits.

tens.

units.

huns.

In the number four hundred and six, there are 6 units of the first order, 0 of the second, and 4 of the third: and in a similar manner we may express, by three figures, any number greater than ninety-nine and less than one thousand. The right hand figure will express units of the first order, the next, units of the second order, and the other, units of the third order.

7. How do you express one hundred? To how many units of the second order is it equal? To how many of the first order? May it be considered a single unit? Of what order is it? How many units of the third order in 200? In 300? In 400? In 500? In 600? In 700? In 800? In 900? Of what is the number 375 composed? The number 406? What numbers may be expressed by three figures? What order of units will each figure express?

8. To express ten units of the third order, or one thousand, we form a new combination by writing three ciphers on the right of 1; thus, 1000

Now, although this thousand expresses one thousand units of the first order, it is, nevertheless, but one single thousand, and may be regarded as a unit of the fourth order.

Proceeding in this manner, we may form as many orders of units as we please: thus, a single unit of the first order is expressed by

a unit of the second by 1 and a 0; thus,

1,

10,

a unit of the third order by 1 and two 0's; thus, 100, a unit of the fourth order by 1 and three O's; thus, 1000, a unit of the fifth order by 1 and four 0's; thus, 10000; and so on, for units of higher orders.

9. We see, from the language of figures,

1st. That the same figure expresses different values according to the place which it occupies.

:

2d. That units of the first order always occupy the place on the right units of the second order the second place from the right: units of the third order the third place; and so on for places still to the left.

3d. That ten units of the first order make one of the second; ten of the second one of the third; ten of the third one of the fourth, and so on for the higher orders.

4th. That when figures are written by the side of each other, ten units in any one place make me unit of the place next to the left.

8. To what are ten units of the third order equal? How do you express them? How is a single unit of the first order expressed? How do you express a unit of the second order? One of the third? One of the fourth? One of the fifth?

9. On what does the value of the same figure depend? What is the unit of the first place on the right? What is the unit of the second place? What is the unit of the third place? Of the fourth? Of the fifth? Sixth? How many units of the first order make one of the second? How many of the second one of the third? How many of the third one of the fourth? &c. When figures are written by the side of each other, how many units in any place make one unit of the place next to the left?

10. Expressing or writing numbers by figures, is called NOTATION. Reading the order of their places, correctly, when written, is called NUMERATION.

4. Write 4 units of the first order, 5 of the second, 6 of the third, and 8 of the fourth.

Ans.

5. Write 9 units of the fifth order, none of the fourth, 8 of the third, 7 of the second, and 6 of the first.

Ans. 90876. 6. Write 1 unit of the sixth order, 5 of the fifth, 4 of the 4th, 9 of the third, 7 of the second, and none of the first.

7. Write 4 units of the 11th order.

8. Write forty units of the second order.

Ans.

Ans. 400.

9. Write 60 units of the third order, with four of the 2d, and 5 of the first.

10. Write 16 units of the 12th order, with 8 of the 9th, 4 of the 5th, 7 of the 2d, and 1 of the 1st.

11. Write 7 units of the ninth order, with 6 of the 7th, 9 of the third, 8 of the 2d, and 9 of the first.

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

13. Write 14 units of the 10th, 6 of the 8th, 7 of the 3d, and 3 of the first.

12th order, with 9 of the 6th, 6 of the 5th, 5 of the

14. Write 13 units of the 13th order, 8 of the 12th, 7 of the 9th, 6 of the 8th, 9 of the 7th, 7 of the 6th, 3 of the fourth, and 9 of the first.

15. Write 9 units of the 18th order, 7 of the 16th, 4 of the 15th, 8 of the 12th, 3 of the 11th, 2 of the 10th, 1 of the 9th, 0 of the 8th, 6 of the 7th, 2 of the third, and 1 of the 1st.

10. What is notation? What is numeration? Which way do you numerate?

.

6, 407, 212,

936,

876,

541

57, 289, 678, 541, 297, 313

920, 323, 842, 768, 319, 675

The words at the head of the numeration table, units, tens, hundreds, &c., are equally applicable to all numbers, and must be committed to memory; after which, the pupil may read the Table.

To make the reading of figures easy, they are often separated into periods of three figures each, counting from the right hand.