33. If I divide eighteen boys into classes of 9, many classes will there be ?

34. 9 are in eighteen how many times?

how

35. A man divided eighteen cents among some boys, giving each boy 6 cents. How many boys were there?

36. 6 are in eighteen how many times?

37. A boy had twenty-four sheets of paper, which he made into writing books, putting 4 sheets into a book. How many books did he make?

33. 4 are in twenty-four how many times?

39. If an acre of ground cost 3 dollars, how much can I buy with eighteen dollars? 3 are in eighteen how many times?

40. If a toy-book cost 3 cents, how many can I buy with twenty one cents? 3 are in twenty one how many

times ?

41. If a pail will hold 4 gallons of water, how many pailfuls will a tub contain, which holds sixteen gallons? 4 are in sixteen how many times ?

42. 4 quarts make a gallon. How many gallons are there in twenty quarts? 4 are in twenty how many times?

43. How many gallons are there in twelve quarts? In 8 quarts? In twenty four quarts? In twenty eight quarts? In thirty two quarts? In thirty six quarts? In forty quarts?

44. 2 pints make a quart. How many quarts are there in 4 pints? In 6 pints? In 8 pints? In ten pints? In twelve pints? In fourteen pints? In sixteen pints? In eighteen pints? In twenty pints? In twenty two pints?

45. If a yard of ribbon cost 3 cents, how many yards will 6 cents buy? Will 9 cents? Will twelve cents? Will fifteen cents? Will eighteen? Will twenty one? Will twenty four? Will twenty seven? Will thirty? Will thirty three? Will thirty six ?

46. How many times 5 in ten? In fifteen?. In twenty? In twenty five? In thirty? In thirty five? In forty? In forty five? In fifty? In fifty five? In sixty?

47. How many times 6 in twelve? In eighteen? In 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? In 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? In 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 adTanced pupils who study this book.

101010

ARITHMETIC.

NUMERATION.

III. For those who omit the preceding exercises,

we here repeat the definition of a unit, § I. ART I.

A SINGLE THing of any kind is called a unit or unity.
You know very well what NUMBERS are.

When you

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 value 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.

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

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

ture may contain several men, so

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

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

ounces.

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.

I

Moreover, suppose I have a barrel of a certain size. 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

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. larger units shall be expressed by the figures 1, 2, 3, &c. and the smaller, by the common figures 1, 2, 3, &c. Then is ten times 1, 2 is ten times 2, &c.

The

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 1s, and each of these ten 1s 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 1s, 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, 2, 3, &c., units of the third order.

THE FIRST ORDER MAY BE CALLED UNITS, SIMPLY; THE SECOND ORDER, 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 examples, whose answers are given. The figures, used in the book, should be imitated.]