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47. How many times 6 in twelve ? In eighteen? În twenty four ? In thirty! In thirty six? In forty two? In forty eight ? In fifty four? In sixty ? In sixty six ? In seventy two?

48. How many times 7 in fourteen ? In twenty one ? In twenty eight ? In thirty five? In forty two? In forty nine? In fifty six ? In sixty three? In seventy ? In seventy seven? In eighty four ? In ninety one ?

49. How many times 8 in sixteen? In twenty four? In thirty two? In forty? In forty eight? In fifty six? In sixty four? In seventy two? In eighty? In eighty eight? In ninety six ? In one hundred and four ?

50. How many times 9 in eighteen? In twenty seven? In thirty six? In forty five ? In fifty four ? În sixty three ? In seventy two? In eighty one? In ninety? In ninety nine? In one hundred and eight? In one hundred and seventeen?

51. How many times ten in twenty ? In thirty? In forty? In fifty ? În sixty? In seventy? In eighty ? In ninety ? In one hundred ? In one hundred and ten ? In one hundred and twenty ? In one hundred and thirty ?

52. How many times eleven in twenty two ? In thirty three ? In forty four ? In fifty five? In sixty six ? In seventy seven? In eighty eight? In ninety nine? In one hundred and ten? In one hundred and twenty one? In one hundred and thirty two?

53. How many times twelve in twenty four? In thirty six? In forty eight? In sixty ? In seventy two? In eighty four? In ninety six ? In one hundred and eight? In one hundred and twenty? In one hundred and thirty two? In one hundred and forty four?

Note. The preceding lessons may be omitted by the more actor maced pupils who study this book.

ARITHMETIC..

NUMERATION.

When you

$ III. For those who omit the preceding exercises, we here repeat the definition of a unit, § 1. Art i.

A SINGLE THING OF ANY KIND IS CALLED A UNIT OR UNITY.

You know very well what NUMBERS are. say, John has three marbles, you tell me what number of marbles John has. When you say, my sister has two oranges, but I have only one, you tell me the number of oranges your sister has, and also the number you have yourself. As you understand this, we will not trouble you at present with a definition of number. The preceding exercises have required you to use numbers; and you will recollect that the names of a few of the first of these are, ONE, TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE ; which are represented by the characters, 1, 2, 3, 4, 5, 6, 7, 8, 9. Besides these, the character 0, is employed, which means nought or nothing; that is, it has no value in itself. It is, however, very useful, as will be seen hereafter. It is usually called Cypher or zero.

We have no single character, to represent any number greater than nine.

Those already given are sufficient, as we shall see, to express any numbers, however large. The first nine are often called digits. The whole are called FIGURES. The first nine are also called siGNIFICANT FIGURES, because they have some ralue or signify something. The cypher is not a significant figure, because it has no value, or signifies nothing. Before we proceed to explain how larger numbers are written, we wish your very diligent attention to the following,

Here is a picture of an auction, or public sale of goods.

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This is a single picture, and it is therefore a vnit.

But there are several men in the picture, and each single man is also a unit. Hence, it appears, that, as one picture may contain several men, so

A UNIT OF ONE KIND MAY CONTAIN SEVERAL UNITS OF ANOTHER KIND. Here is a basket of flowers,

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This is a single basket, and is therefore a unit. But at contains several flowers, and each flower is likewise u unit.

Here we see, exactly as before, that a unit of one kind may contain several units of another.

We have similar examples in measures. One gallon contains four quarts ; that is, one unit of one kind contains four units of another kind. One yard contains three feet; that is, one unit of one kind contains three units of another kind. So, likewise in weights. One pound contains sixteen

One hundred weight contains four quarters, &c. So likewise in coins. One dollar contains one hundred cents. One eagle contains ten dollars, &c.

Moreover, suppose I have a barrel of a certain size. I may employ a man to make another, as many times larger as I please. Thus, as in the picture, I may have one made three times larger. Hence, we see, that

ounces.

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A UNIT OF ONE KIND MAY BE MADE TO CONTAIN AS MANY UNITS OF ANOTHER KIND AS WE CHOOSE,

In like manner, if I draw on paper, a short line, I may easily draw another, ten times as long.

Here, then, I make a unit of the latter kind, ten times as great, as one of the former. But I might have made it nine times, or eleven times, or twelve times, or any number of times as great.

Now, to come to the point at which we are aiming, we have only to apply this to numbers : for, it is plain, that, in numbers, as well as in any thing else, we may have one kind of units, which shall contain several of another kind.

Suppose, then, in numbers, we make one unit of a larger kind equal to ten of a smaller.

This is the way, in which numbers are actually reckoned. Now, as the characters. 1, 2, &c. stand for units of any kind, we may use them to stand for units of the larger kind, as well as for those of the smaller, if we can contrive any way of distinguishing one kind from the other. This we will do at present, by using different kinds of_type. The larger units shall be expressed by the figures 1, 2, 3, &c. and the smaller, by the common figures 1, 2, 3, &c. Then 1 is ten times 1, 2 is ten times 2, &c.

In the same manner, we may have a different kind of figures still, as 1, 2, 3, &c. to stand for units of a larger kind than either of these ; each of which shall be equal to ten of the kind denoted by the figures 1, 2, 3, &c. Then, since 1 stands for ten times 1, or ten Is, and each of these ten ls stands for ten 1s, 1 is equal to ten times ten 1s, or one hundred 1s.

We might have another kind still, each of which should be equal to ten Is, and of course equal to one thousand 1s, and so on. Here we have several kinds of units. For convenience, it is best to call the smallest, represented by 1, 2, 3, &c. units of the first order, the next, represented by 1, 2, 3, &c., units of the second order, and those represented by 1, 9, 8, &c., units of the third order.

TAÐ FIRST ORDER MAY BE CALLED UNITS, SIMPLY ; THE SECOND OR. DER, TENS ; AND THE THIRD ORDER, HUNDREDS ; TAKING THEIR NAMES

FROM THEIR VALUES.

(Let the pupil be now required to write the following numbers, placing units on the right; tens, next towards the left; and hundreds, next; as in the exam. ples, whose answers are given. The figures, used in the book, should be imitated.}

Write one unit of the second order, (or one ten,) and one unit of the first order, (or one unit.)

Ans. 11. Write one unit of the third order, (or one hundred,) and one ten, and one unit.

Ans. 111. Write one hundred, one ten, and five units. Ans. 1 15. Write two hundreds, three tens, and seven units. Ans. 237. Three hundreds, five lens, and eight units.

Ans. 358. Two hundreds, eight tens, and nine units. Seven hundreds, six tens, and three units. Two tens, and two units. Nine tens, and six units. Four hundreds, and six tens, and four units. Five hun. dreds, five tens, and five units. Nine hundreds, seven tens, and three units. Four hundreds, eight tens, and four units. Eight hundreds, and nine tens, and nine units. Two hundreds, six tens, and three units. One hundreds, tw, tens, and three units. Two hundreds, five tens, and seven units. One ten, and three units, Seven tens, and three units. Nine hundreds, nine tens, and nine units.

In all these examples, units of the first order, come first on the right; those of the second come second ; those of the third come third ; so that if the same kind of figures were used for all, we should know the several orders, by their places.

Where do units of the first order come ? Where do units of the second order come ? Where do those of the third order come? Then,

THE NUMBER OR NAME OF ANY ORDEK, IS THE SAME WITH THE NUMBER OR NAME OF ITS PLACE, COUNTING FROM THE

RIGHT.

If we used the same figures for all the orders, how would

you know the order to which any figure belonged ? Then, using the common figures, write the following numbers.

One ten, and one unit, or eleven. Ans.' 11.
One ten, and two units, or twelve.

12.
One ten, and three units, or thirteen. 13.
Fourteen, or one ten, and four units. 14.
Fifteen, or one ten, and five units.

15.
Sixteen, or one ten, and six units.

16. Seventeen, or one ten, and seven units. 17. Eighteen, or one ten, and eight units. 18.

Nineteen, or one ten, and nine units. 19. One hundred and thirty-four, or one hundred, and three tens, and four units. Ans. 134.

Seven hundred and sixty-three, or seven hundreds, and six tens, and three units.

Now write one ten, or one ten and no units. Ans. 10. Here we see the use of the 0, or cypher ; for it we were to write nothing but 1, it would read one unit of the first order, or one unit simply ;

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