8. It will be perceived from the foregoing table, that the same figures, standing in different places, have different values.

When they stand alone or in the right hand place, they express units or ones, which are called units of the first order.

When they stand in the second place, they express tens, which are called units of the second order.

When they stand in the third place, they express hundreds, which are called units of the third order.

When they stand in the fourth place, they express thousands, which are called units of the fourth order, &c.

For example, the figures 2, 3, 4, and 5, when arranged thus, 2345, denote 2 thousands, 3 hundreds, 4 tens, and 5 units; when arranged thus, 5432, they denote 5 thousands, 4 hundreds, 3 tens, and 2 units.

9. Ten units make one ten, ten tens make one hundred, and ten hundreds make one thousand, &c.; that is, ten of any lower order, are equal to one in the next higher order Hence, universally,

10. Numbers increase from right to left in a tenfold ratio; that is, each removal of a figure one place towards the left, increases its value ten times.

11. The different values which the same figures have, are called simple and local values.

The simple value of a figure is the value which it expresses when it stands alone, or in the right hand place.

QUEST.-8. Do the same figures always have the same value? When standing alone or in the right hand place, what do they express? What do they express when standing in the second place? In the third place? In the fourth? 9. How many units make one ten? How many tens make a hundred? How many hundreds make a thousand? Generally, how many of any lower order are required to make one of the next higher order? 10. What is the general law by which numbers increase? What is the effect upon the value of a figure to remove it one place towards the left? 11. What are the differ ent values of the same figure called? What is the simple value of a figure? What the local value?

The simple value of a figure, therefore, is the number which its name denotes.

The local value of a figure is the increased value which it expresses by having other figures placed on its right. Hence, the local value of a figure depends on its locality, or the place which it occupies in relation to other numbers with which it is connected. (Art. 8.)

OBS. This system of notation is also called the decimal system, because numbers increase in a tenfold ratio. The term decimal is derived from the Latin word decem, which signifies ten.

NUMERATION.

12. The art of reading numbers when expressed by figures, is called Numeration.

13. The different orders of numbers are divided into periods of three figures each, beginning at the right hand.

QUEST.-Upon what does the local value of a figure depend? Obs. What is this system of notation sometimes called? Why? 12. What is Numeration? Repeat the numeration table, beginning at the right hand. What is the first place on the right called? The second place? The third? Fourth? Fifth Sixth? Seventh? Eighth? Ninth? Tenth, &c.? 13. How are the orders of numbers divided?

The first, or right hand period is occupied by units, tens, hundreds, and is called units' period; the second is occupied by thousands, tens of thousands, hundreds of thousands, and is called thousands' period, &c.

The figures in the table are read thus: One hundred and twenty-three trillions, eight hundred and sixty-one billions, five hundred and eighteen millions, nine hundred and twenty-four thousand, two hundred and sixty-three.

14. To read numbers which are expressed by figures.

Point them off into periods of three figures each; then, beginning at the left hand, read the figures of each period as though it stood alone, and to the last figure of each, add the name of the period.

OBS. 1. The learner must be careful, in pointing off figures, always to begin at the right hand; and in reading them, to begin at the left hand.

2. Since the figures in the first or right hand period always denote units, the name of the period is not pronounced. Hence, in reading figures, when no period is mentioned, it is always understood to be the right hand, or units' period.

EXERCISES IN NUMERATION.

Note.-At first the pupil sbould be required to apply to each figure the name of the place which it occupies. Thus, beginning at the right hand, he should say, “Units, tens, hundreds,” &c., and point at the same time to the figure standing in the place which he mentions. It will be a profitable exercise for young scholars to write the examples upon their slates or paper, then point them off into periods, and read them.

QUEST.-What is the first period called? By what is it occupied? What is the second period called? By what occupied? What is the third period called? By what occupied? What is the fourth called? By what occupied ? What is the fifth called? By what occupied? 14. How do you read numbers expressed by figures? Obs. Where begin to point them off? Where to read them? Do you pronounce the name of the right hand period? When no period is named, what is understood?

EXERCISES IN NOTATION.

15. To express numbers by figures.

Begin at the left hand of the highest period, and write the figures of each period as though it stood alone.

If any intervening order, or period is omitted in the given number, write ciphers in its place.

Write the following numbers in figures upon the slate or black-board.

1. Sixteen, seventeen, eighteen, nineteen, twenty. 2. Twenty-three, twenty-five, thirty, thirty-three.

3. Forty-nine, fifty-one, sixty, seventy-four.

4. Eighty-six, ninety-three, ninety-seven, a hundred.

QUEST.-15. How are numbers expressed by figures? If any intervening order is omitted in the example, how is its place supplied?

5. One hundred and ten.

6. Two hundred and thirty-five.
7. Three hundred and sixty.

8. Two hundred and seven.
9. Four hundred and eighty-one.

10. Six hundred and ninety-seven.

11. One thousand, two hundred and sixty-three. 12. Four thousand, seven hundred and ninety-nine. 13. Sixty-five thousand and three hundred.

14. One hundred and twelve thousand, six hundred and seventy-three.

15. Three hundred and forty thousand, four hundred and eighty-five.

16. Two millions, five hundred and sixty thousand.

17. Eight millions, two hundred and five thousand, three hundred and forty-five.

18. Ten millions, five hundred thousand, six hundred and ninety-five.

19. Seventeen millions, six hundred and forty-five thousand, two hundred and six.

20. Forty-one millions, six hundred and twenty thousand, one hundred and twenty-six.

21. Twenty-two millions, six hundred thousand, one hundred and forty-seven.

22. Three hundred and sixty millions, nine hundred and fifty thousand, two hundred and seventy.

23. Five billions, six hundred and twenty-one millions, seven hundred and forty-seven thousand, nine hundred and fifty-four.

24. Thirty-seven trillions, four hundred and sixty-three billions, two hundred and ninety-four thousand, five hundred and seventy-two.