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ARITHMETIC.

INTRODUCTION.

1. EVERY THING which admits of increase or diminution is called quantity.

II. Quantities are made up of parts. Thus, six apples is a quantity which is made up of six single apples; ten trees is a quantity made up of ten in. dividual trees; and fifteen peaches is a quantity made up of fifteen single peaches,

III. In the classes of natural objects, such as horses, sheep, apples, trees, peaches, &c. one of the species, or kind, is called the unit of the quantity. Thus, one apple is the unit of the quantity six apples; one tree is the unit of the quantity ten trees, and one peach the unit of the quantity fifteen peaches.

IV. There are other quantities, besides the classes of natural objects, which are also made up of parts. For example, ten feet in length is a quantity which is made up of ten separate feet; six pounds of tea is a quantity made up of six separate parts of one pound each, and four hours of time is a quantity made up of four separate hours.

V. We can conceive of a quantity only by comparing it with another quantity of the same kind, whose value is fixed and known. Thus, when we speak of the distance ten feet, we are only able to conceive of it, bý first knowing the length of one foot, and then understanding that this length may be laid off ten times in the given distance ten feet.

When we speak of six pounds of tea, we understand th:

one hour.

it is such a quantity as comes from mixing six separate pounds together. When we speak of four hours of time, we mean a duration four times longer than

VI. In each of the above cases we compare the quantity spoken of with a quantity of the same kind, whose value is known. This known value is called the unit of the quantity, and is one of the equal parts of which the quantity is made up.

VII. In the examples above, one foot is the unit of the distance ten feet; one pound of tea is the unit of the quantity six pounds of tea, and one hour the unit of the quantity four hours of time.

VIII. NUMBERS are the expressions for several things of the same kind: and they show how many times a quantity contains its unit. Thus, when we say three trees, the number three expresses three things of the same kind: or if we say six pounds of tea, we understand that the quantity six pounds of tea contains its unit one pound, six times.

IX. In numbers, one is the most simple unit.

QUESTIONS. 1. What is quantity ?

II. How are quantities made up? Of what is the quantity Bix apples made up ?-ten trees ?-fifteen peaches ?

III. In the classes of natural objects what is one called ? What is the unit of six apples ?-Of fifteen poacles ?

IV. Of what is the quantity ten feet made up ?- Six pounds of tea ?-Four hours of time?

V. How do we conceive of a quantity ?-How do we con. ceive of ten feet? Of six pounds of tea ?-Of four hours of time ?

VI. What is the known value called? What is it?

VII. What is the unit of the distance ten feet 3-of six pounds of tea ?-Of four hours of time ? VITI. What are numbers ? What do they show ?'

What is the most simple unit?

Two,

ARITHMETIC. $ 1. ARITHMETIC treats of numbers. It is both a science and an art.

It is a science, when it explains the nature and properties of numbers; and an art, when it treats of the best methods of using them.

2. In Arithmetic, numbers are expressed by cer. tain characters, called figures.* There are ten of these characters. They are

O'which is called a cipher, or Naught,
1

One,
2
3

Three,
4

Four, 5

Five, 6

Six, 7

Seven, 8.

Eight, 9

Nine. * Note..Thero is another method called the Roman, in which, numbers are represented by letters ; I represents one, V five, X ten, L fifty, C one hundred, D five hundred, &c.

TABLE. One I.

Sixty

LX. Two II.

Seventy LXX. Threo III.

Eighty

LXXX Four IIII. or IV. Ninety LXXXX, or XC. Five V.

One hundred C. Six VI.

Two hundred CC.
Seven VII.

Three hundred CCC.
Eight
VIII.

Four hundrod CCCC.
Nine VIIII. or IX. Five hundred D.
Ten X.

Six hundrod DC.
Twenty XX.

Seven hundred DCC.
Thirty XXX.

Eight hundred DCCC.
Forty XXXX. or XL. Nine hundred DCCCC.
Fifty
L.

One thousand M.

11

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Ø 3. The character O is used to denote the absence of a thing As, if we wished to express by figures that there were no apples in a basket, we should write, the number of apples in the basket is 0. The nine other figures are called, significant figures. 1 expresses a single thing, or a unit of a quantity. 2

two things of the same kind, or two units. 3 three things

or three units. 4. four things

or four units. 5 five things

or five units. 6 six things

or six units. 7 seven things

or seven units. eight things

or eight units. nine things

or nine units. Ø 4. If we wish to express the number ten, we have no separate character for it. We must combine the characters already known. This'we do by writing 0 on the right hand of the 1. Thus, 10 which is read

Ten. This 10 is equal to ten of the units expressed by 1. It is, however, but a single ten, and in this sense may be regarded as a unit, the value of which is ten times greater than the unit expressed by 1.

05 When two figures are thus written by the side of each other, the one on the right is called the place of units, the other, the place of tens, or units of the second order ; and each unit of the second order is equal to ten units of the first order.

16. When units simply are mentioned, units of the first order are always meant : Two tens, or twenty, are written

20 Three tens, or thirty,

30 Four tens, or forty,

40 Five teng, or fifty,

50 Six tens, or sixty,

60 Soven tens, or seventy,

70 Eight tens, or eighty,

80 Nine tens, or ninety,

90

7. The intermediate numbers between 10 and 20, between 20 and 30, &c. may readily be expressed by considering the tens and units of which they are composed. For example, the number twelve is composed of one ten and two units. It must therefore be written by writing 1 in the place of tens, and 2 in the place of units ; thus,

12 Eighteen has 1 ten and 8 units, and is written 18 Twenty five has 3 tens and five units, and is written 25 Thirty seven has 3 tens 7 units, and is written 37 Fifty four has 5 tens and 4 units, and is written 54 Eighty nine has 8 tens and 9 units, and is written 89 Ninety nine has 9 tens and 9 units, and is written 99

98. In order to express one hundred, or ten units of the second order, we have to form a new combination. It is done thus,

100 by writing two ciphers on the right of 1. This number is read, one hundred. Now this one hundred expresses 10 units of the second order, or one huna dred units of the first order. Bụt the one hundred is but an individual hundred, and in this light may be regarded as a unit of the third order.

$ 9. With these combinations all the numbers between 1 and one thousand

may

be expressed, For example, in the number three hundred and seventy five, there are 3 hundreds, 7 tens, and 5 units. We therefore, to express 3 units of the 3d order, 7 units of the second order, and 5 of the 1st. Hence we write

37 5 In the number eight hundred and ninety nine, there are 8 units of the 3d order, 9 of the 2d, and 9 of the 1st.

It is written

are,

w buns.
v tens
er units

huns.
co tens
co units

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