30. Since the number expressed by any figure depends upon the place it occupies, it follows that figures have two values, Simple and Local.

31. The Simple Value of a figure is its value when taken alone;

Thus, 4, 7, 2.

32. The Local Value of a figure is its value when used with another figure or figures in the same number.

Thus, in 325, the local value of the 3 is 300, of the 2 is 20, and of the 5 is 5 units.

When a figure occupies units' place, its simple and local values are the same.

33. The leading principles upon which the Arabic notation is founded are embraced in the following laws:

GENERAL LAWS.

I. All numbers are expressed by applying the ten figures to the different orders of units.

II. The different orders of units increase from right to left, and decrease from left to right, in a tenfold ratio.

III. Every removal of a figure one place to the left, increases its local value tenfold; and every removal of a figure one place to the right, diminishes its local value tenfold.

From this analysis of the principles of Notation and Numeration, we derive the following rules:

Rule for Notation. I. Beginning at the left hand, write the figures belonging to the highest period.

II. Write the hundreds, tens, and units, of each successive period in their order, placing a cipher wherever an order of units is omitted.

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

II. Beginning at the left hand, read each period separately, and give the name to each period, except the last, or period of units.

34. Until the pupil can write numbers readily, it may be well for him to write several periods of ciphers, point them off, over each period write its name, and then write the given numbers underneath, in their appropriate places.

Thus,

000, 000, 000.

Millions, Thousands, Units.

EXERCISES IN NOTATION AND NUMERATION.

Express the following numbers by figures:

1. Four hundred thirty-six.

2. Seven thousand, one hundred sixty-four.
3. Twenty-six thousand, twenty-six.
4. Fourteen thousand, two hundred eighty.
5. One hundred seventy-six thousand.
6. Four hundred fifty thousand, thirty-nine.

7. Four hundred eighty-three million, eight hundred

sixteen thousand, one hundred forty-nine.

8. Ninety-five million.

9. Nine hundred thousand, ninety.

10. Ten million, ten thousand, ten hundred, ten.

Point off, numerate, and read the following num

23. Write seven million thirty-six.

24. Write five hundred sixty-three thousand, four. 25. Write one million, ninety-six thousand.

26. A certain number contains 3 units of the seventh order, 6 of the fifth, 4 of the fourth, 1 of the third, 5 of the second, and 2 of the first; what is the number?

27. What orders of units are contained in the number 290648?

ADDITION.

35. Addition is the process of uniting several numbers of the same kind into one equivalent number.

36. The Sum or Amount is the result obtained.

ORAL EXERCISES.

1. A farmer paid 6 dollars for a straw-cutter, and 9 dollars for a plow. What did he pay for both?

SOLUTION. - He paid the sum of 6 dollars and 9 dollars, which is 15 dollars.

2. John gave 4 apples to James, 8 to Henry, and 9 to Fred. How many did he give to all?

3. I gave 7 dollars for a barrel of flour, 5 dollars for a hundred-weight of sugar, and 6 dollars for a tub of butter. What did I give for the whole?

4. I have two pear trees; last year one tree produced 12 bushels of pears, and the other 11 bushels. How many bushels did both produce?

5. A man bought 4 cords of wood for 12 dollars, and 7 bushels of corn for 5 dollars. What did he pay for both? 6. James gave 11 cents for a slate, and had 8 cents left. How many cents had he at first?

7. A lady paid 5 dollars for a bonnet, 10 dollars for a shawl, and had 7 dollars left. How much money had she at first?

8. In a shop there are 8 men, 9 boys, and 6 girls at work. How many persons are at work in the shop?

9. Robert bought a quire of paper for 12 cents, and a slate for 13 cents; he gave 10 cents to his sister. How much money did he pay out in all ?

10. A man bought 4 bushels of wheat for 7 dollars, 18 bushels of corn for 11 dollars, and 2 cords of wood for 5 dollars. What did he pay for the whole?

11. A farmer has 6 cows in one yard, 9 in another, and as many in the third yard as in both the others. How many cows has he?

12. There were 7 desks on the first row of a schoolroom, 7 desks on the second row, and 6 on the third row. How many desks were there in the three rows?

13. Mary is 8 years old, Jane is 3 years older than Mary, and Anna is 4 years older than Jane. How old is Jane? How old is Anna?

14. A little boy had 9 cents in his bank, his aunt gave him 5 cents, and his uncle gave him 10 cents. did he then have in the bank?

How much

15. George has 8 marbles in his pocket, 4 in his right hand, and 5 in his left hand. How many marbles has he altogether?

37. The Sign of Addition is the perpendicular cross, +, called plus. It shows that the numbers connected by it are to be added.

Thus, 3+ 4+ 7, read 3 plus 4 plus 7, or 3 and 4 and 7.

38. The Sign of Equality is two short, parallel, horizontal lines =. It shows that the numbers, or combination of numbers, connected by it are equal.

Thus, 4+89+ 3, read the sum of 4 and 8 is equal to the sum of 9 and 3.