The local value of a figure depends on its locality, or the place which it occupies in relation to other figures with which it is connected, counting from the right. (Art. 8.)

OBS. 1. The reason for assigning different values to the same figures according to the place which they occupy, is to enable us to express large numbers intelligibly, and at the same time with few characters. Otherwise we must have as. many different characters as we have different numbers to express, and the labor of learning them would be greater than that of learning the whole Eng lish language.

2. The Arabic notation is also called the decimal notation, because its orders increase in a tenfold ratio. The term decimal is derived from the Latin word decem, which signifies ten.

NUMERATION.

11. NUMERATION is the art of reading numbers expressed by figures.

OBS. Numeration bears the same relation to Notation, that reading does to writing; though often confounded, they are entirely distinct.

The pupil has already become acquainted with the names of numbers, from one to a thousand. He will now easily learn to read and express the higher numbers in common use, from following scheme, called the

the

QUEST.-Upon what does the local value of a figure depend? Obs. What is the Arabic notation sometimes called? Why? 11. What is Numeration? Repeat the Numeration Table, beginning at the right hand. What occupies the Fourth? Fifth ? first place on the right? The second place? The third? Sixth? Seventh? Eighth? Ninth? Tenth, &c.?

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

The first period on the right, is called units' period, because it is occupied by units, tens of units, and hundreds of units.

The second is called thousands' period, because it is occupied by thousands, tens of thousands, and hundreds of thousands, as may be seen from the table above.

The figures in the table are read thus: Five hundred and sixty-eight quadrillions, three hundred and forty-two trillions, nine hundred and seventy-five billions, eight hundred and ninety-seven millions, six hundred and forty-five thousand, four hundred and thirty-two. Hence,

13. To read numbers expressed by figures.

First, point them off into periods of three figures each, counting from the right.

Then, beginning at the left hand, read the figures of every period as though it stood alone, and to the last figure of each, add the name of the period.

OBS. 1. In pointing off figures, the learner must be careful to begin at the right band; 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.

14. The method of dividing numbers into periods of three figures, as in the preceding articles, is called the French Numeration, because it was invented by the French.

The English divide numbers into periods of six figures. The French method is the more simple and convenient. It is generally used on the continent of Europe, as well as in America, and has been recently adopted by some English authors.

What is the first

QUEST.-12. How are the orders of numbers divided? period called? Why? What is the second called? Why? What is the third called? Why? What is the fourth called? Why? What is the fifth called? Why? 13. How do you read numbers expressed by figures? Do you pronounce the name of the right hand period? When no period is named, what is understood? 14. In the French numeration, how many figures are there in a period? How many in the English method? Which method is preferable?

EXERCISES IN NUMERATION.

Note. In reading large numbers, beginners should at first point to each figure, and pronounce its name. Thus, beginning at the right hand, he should say, "Units, tens, hundreds, thousands," &c. It will be a profitable exercise for young scholars to write the examples upon their slates or paper, then point shem off into periods, and read them.

Read the numbers expressed by the following figures:

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 :

QUEST.-15. How do you express numbers by figures? If any intervening order, or period is omitted in the given number, how is its place supplied?

6. One thousand, and forty-two.

7. Thirty thousand, nine hundred and seven. 8. Forty-six thousand, and four hundred.

9. Ninety-two thousand, one hundred and eight. 10. Sixty-eight thousand, and seventy.

11. One hundred and twenty-four thousand, six hundred and thirty.

12. Two hundred thousand, one hundred and sixty.

13. Four hundred and five thousand, and forty-five.

14. Three hundred and forty thousand.

15. Nine hundred thousand, seven hundred and twenty. 16. One million, and seven hundred thousand.

17. Thirty-six millions, twenty thousand, one hundred and fifty.

18. One hundred millions, and forty-five.

19. Mercury is thirty-seven millions of miles from the sun. 20. Venus, sixty-nine millions.

21. The Earth, ninety-five millions.

22. Mars, one hundred and forty-five millions.

23. Jupiter, four hundred and ninety-four millions.

24. Saturn, nine hundred and seven millions.

25. Herschel, one billion, eight hundred and ten millions. 26. Seven billions, nine hundred millions, and forty thousand. 27. Sixty billions, seven millions, and four hundred.

28. One hundred and thirteen billions, six hundred and fifty thousand.

29. Four hundred and six billions, eighty millions, and seven hundred.

30. Twenty-five trillions, and ten thousand.

31. Two hundred and six billions, five hundred and sixty thousand, and forty-five.

32. Six hundred millions, seventeen thousand, three hundred and eight.

33. Ninety-seven trillions, sixteen millions, seventy thousand, and thirty.

34. Eight hundred and forty billions, fifty millions, three hundred and one thousand.

35. Three hundred and sixty-five quadrillions, two hundred trillions, six hundred and ninety billions, seven millions, three thousand and six.

SECTION II.

ADDITION.

MENTAL EXERCISES.

ART. 16. Ex. 1. George bought a slate for 9 cents, a sponge for 6 cents, and a pencil for 1 cent: how many cents did he pay for all?

Solution.-9 cents and 6 cents are 15 cents, and 1 cent more makes 16 cents. He therefore paid 16 cents for all.

1 are 31 are 4 1 are 5 1 are 61 are 7 1 are 8

2 " 4 2"

3 " 5 3 "

52"
63"

4 66

64"

74"

62 "
7 3 66
8 4

7 26

66

9

5 4

6 66

75"
86"

85" 9 5 "10

1 are 91 are 10 81 2 " 9 2 10 2 66 11 8 3" 9 310 311 312 4" 10 4 6 11 4 12 4" 13 5" 11 512 5" 13 5" 141 6 "14 6" 15 7 "15 716) 15 8 169 17 10

96 66 10 6 11 6 " 12 6 4 13 7 " 9 7 10 7 11 7 66 127" 13 7 14 8" 10 8" 118 "128" 13 8" 14 8" 9 11 9" 12 9 64 13 9 149 15 9 10 "12 10" 13 10" 14 10 15 10 16 10

Note. It is indispensable to accuracy both in arithmetic and business, to have ,he common arithmetical tables distinctly and indelibly fixed in the mind. Great care should therefore be taken to prevent them from being recited mechanically, or from a knowledge of the regular increase of numbers.

11. How many are 12 and 10? 22 and 10? 32 and 10? 42 and 10? 52 and 10? 62 and 10? 72 and 10? 82 and 10? 92 and 10?

12. How many are 24 and 10? 36 and 10? 48 and 10? 53 and 10? 67 and 10? 91 and 10? 86 and 10? 78 and 10? 69 and 10 97 and 10?

13. How many are 19 and 4? 29 and 4? 39 and 4? 79 and 4? 59 and 4? 89 and 4? 99 and 4? 69 and 4? 49 and 4?

14. How many are 17 and 8? 27 and 8? 47 and 8? 67 and 8? 57 and 8? 97 and 8? 87 and 8?

15. How many are 16 and 7? 26 and 7? 56 and 7? 86 and 7? 76 and 7? 96 and 7?

16. How many are 14 and 6? 24 and 6? 84 and 6? 74 and 6? 54 and 6? 64 and 6? 94 and 6?