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in questioning him on the meaning of each sentence, which he may be required to read, will be of incalculable advantage.

When pupils shall have been taught how to study, let them be required to get their lessons, and recite them. If the present book is not thought by teachers to contain a sufficient description, and a sufficient explanation of everything, let them try to find one that does, for if pupils present themselves before the blackboard at the time of recitation, with the expectation that the teacher is to explain to the class, and help them through with what they cannot go through themselves, they will not feel that they must have studied themselves; and the paltry oralizing of the teacher will not be listened to, or if heard, will not be understood, or at best, not retained in memory. Pupils may be made to see things for the moment, while no abiding impression will remain on their minds. They will often proceed, in such a manner, through a book, and perhaps have the mistaken idea that they understand its contents

to perpetuate the evil of superficialism, perhaps, themselves, as teachers. Pupils will never have a sufficient understanding of a subject till they shall have studied it carefully themselves, and mastered each part by severe personal application.

Recitation by analysis will be found more conducive to thorough scholarship than adherence to any written questions. Let the class, or any member of the class, be able to commence at the beginning and go through with the entire lesson without any suggestion from the teacher, - a thing that is perfectly practicable and easily attainable. Let pupils be called on, at the pleasure of the teacher, in any part of the class, to go on with the recitation, even to proceed with it in the midst of a subject, the topic in no case ever being named by the teacher. They will thereby become accustomed to give their attention to the recitation, and they will be profited from it, besides securing habits of attention, which will be of incalculable value.

In fine, let arithmetic be studied properly, and more valuable mental discipline will be acquired from it, than is often attained from the whole course in mathematics usually assigned by college faculties. It is not the extent, but the value of acquisitions in mathematics, which is desirable.

W. 6. B.

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SIMPLE NUMBERS. Notation and Numeration, 9 Contractions in Division,

59 Addition, 16 Review of Division,

.63 Review of Numeration and Addition, 22 Miscellaneous Exercises,

65 Subtraction,

23 Problems in the Measurement of Rec. Review of Subtraction,

29
tangles and Solids,

69 Multiplication,

31

illustration by Diagram,71 illustration by Diagram, 34 Definitions,

73 Contractions in Multiplication, 40 General Principles of Division,

74 Review of Multiplication, 45 | Cancelation,

75 Division, 47 Common Divisor,

73 illustration by Diagram, ..50 Greatest Common Divisor,

78

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COMMON FRACTIONS. Notation of Common Fractions, . 80 Multiplication of whole numbers by a Proper, Improper, &c., 82 fraction,

95 Reduction of Fractions,

83 | Multiplication of one fraction by To reduce a fraction to its lowest

another,

96 terms, 85 General Rule,

97 Addition and Subtraction of Fractions, . 87 Examples in Cancelation,

98 Common Denominator, 87 | Division of Fractions,

99 Ist method, 88

by a whole num2d method, 89 ber, two ways,

100 Least Common Denominator, or Least Division of whole numbers by a fraction, 101 Common Multiple, 90 Division of one fraction by another,

103 New Numerators, . 91 General Rule,

103 General Rule,

. 91 Reduction of Complex to Simple FracMultiplication of Fractions, 93 tions,

104 by a whole

Promiscuous Examples, number, two ways, 94 Review of Common Fractions,

106

. 106

DECIMAL FRACTIONS AND FEDERAL MONEY. Decimal Fractions,

108 | Addition and Subtraction of Decimal Notation of Decimal Fractions, 110 Fractions,

119 Table,

111 Addition and Subtraction of Federal To read Decimals, 112 Money,

120 To write Decimals,

112 Multiplication of Decimal Fractions, 121 Reduction of Decimal Fractions, 113

illustration by Diagram, 122 of Common to Decimal Frac

'of Federal Money, tions,

114 Division of Decimal Fractions, Federal Money, .

of Federal Money, Reduction of Federal Money, . 118 Review of Decimal Fractions,

127 Bills,

129

.

123 . 124 126

. 116

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COMPOUND NUMBERS.

131

. 138 . 139

• 132

Definition,

MEASURE OF EXTENSION.
Reduction of Compound Numbers, . . 132 Reduction of, I. Linear Measure,
English Money,

Cloth Measure,
WEIGHT.

II. Land, or Square 1. Avoirdupoise Weight, 135

Measure, :

140 II. Troy Weight,

III. Cubic Measure,

141 II. Apothecary's weight, 137

• 136

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..143

171

. 172

.

. 178
180

· 149

185

MEASURE OF CAPACITY.

Subtraction of Fractional Compound
Reduction of, I. Wine Measure,

Numbers,

164
II. Beer Measure, . 141 Multiplication and Division of Com-
III. Dry Measure,

141
pound Numbers,

165
Time,

. 145 Difference in longitude and time be-
Circular Measure, 146 tween different places. Diagram,
Miscellaneous Table,

147 Review of Con pound Numbers,
Reduction of Fractional Compound Analysis,

. 174
Numbers,

147 Given, price of unity, the quantity, to
To reduce a fraction of a higher de- find the price of quantity,

175
nomination to one of a lower, . 148 Given, quantity, price of quantity, to
To reduce a fraction of a lower to a find the price of unity,

175
higher denomination

: 148 Given, price of unity, price of quantity,
To reduce a fraction of a higher to in- to find the quantity,

175
tegers of a lower denomination, 149 Practice. Aliquot Parts,
To reduce integers of a lower to frac- Articles sold by 100, :
tions of a higher denomination,

by the ton of 2000 lbs.,

181
Reduction of Decimal Compound Num-

price, aliquot part of a pound,
bers, :
151 &c.,

183
Review of Reduction of Compound Articles, quantity less than unity, 184
Numbers,

153 To reduce shillings, pence, &c., to the
Addition of Compound Numbers, 156 decimal of a pound,

Fractional Compound Num- To reduce the decimal of a pound to
bers,

160

shillings, pence and farthings,
Subtraction of Compound Numbers, . 160

PERCENTAGE.
Definition, 187. Rule,
188 | Discount, 215. Commission,

216
Insurance,

190 Time, raie, interest, to find the princi-
Mutual Insurance,

191
pal,

217
Stocks,

193 Principal, interest, time, to find the rate, 218
Brokerage,

191 Principal, rate, interest, to find the
Profit and Loss,
194 time,

. 218
Interest, 195. General Rule,
199 Percentage to find the rate,

.219
Easy way of casting interest when the Bankruptcy,

.220
rate is 6 per cent.,
200 General Average,

221
To compute interest on pounds, shil- Partnership,

222
lings, pence, &c.,

205
on Time,

223
To compute interest when partial pay- Banking,

224
ments have been made,
205 Taxes, method of assessing,

225
Compound Interest,
. 209 Duties,

.227
Table,
Specific,

223
Annual Interest,

Ad Valorem,
Time, rate, and amount given, to find Review of Percentage,

230
the principal,
214) Equation of Payments,

233

... 187

.211
. 212

229

Ratio,
235 | Extraction of the Square Root,

262
In veried and Direct Ratios,
235 Practical Exercises,

266
Compound Ratio,
236 Extraction of the Cube Root,

269
Proportion,
236 Practical Exercises,

273
Rule of Three,
237 Review,

274
To invert both Ratios,
239 | Arithmetical Progression,

275
one Ratio,

239 Simple Interest by Progression, 277
To find the fourth térm of a proportion Annuities by Arithmetical Progression, 279
when three are given,
239 Geometrical Progression,

281
Cancelation Applied,

210 Compound Interest by Progression, 283
Compound Proportion,
.242 Compound Discomt.--Table,

285
Rule,
244 Annuities al Compound Interest,

288
Review of Proportion, .

245 Present worth of Annuities at Com-
Alligation Medial,

246
pound Interest,

289
Alternate,

217 Present worth of Annuities. Table, .290
Exchange,

251

in Rever-
with England,

253
sion,

291
France,
254 Perpetual Annuities,

292
Value of Gold Coins,
255 Permutation,

293
Duodecimals,

256 Miscellaneous Examples,
scale for taking Dimen- Measurement of Surfaces,

298
sions in feet and Decimals of a foot, 259

Solids,

300
Involution,
260 Guaging,

301
Evolution,

. 262 Forms of Notes, &c.,
Bills,

304

. 294

. 303

ARITHMETIC.

NOTATION AND NUMERATION.

1. A single thing, as a dollar, a horse, a man, &c., is called a unit, or one. One and one more are called two, two and one more are called three, and so on. Words expressing how many (as one, two, three, &c.) are called numbers.

This way of expressing numbers by words would be very slow and tedious in doing business. Hence two shorter methods have been devised. Of these, one is called the Roman* method, by letters; thus, I represents one; V, five; X, ten, &c., as shown in the note at the bottom of the page.

The other is called the Arabic method, by certain characters, called figures. This is that in general use.

* In the Roman method, by letters, I represents one; V, five; X, ten; L, fifty; C, one hundred ; D, five hundred; and M, one thousand.

As often as any letter is repeated, so many times its value is repeated, unless it be a letter representing a less number placed before one representing a greater ; then the less number is taken from the greater ; thus, IỂ represents four ; IX, nine, &c., as will be seen in the following

TABLE.
One
I.
Ninety

LXXXX. or XC.
Two
II.

One hundred C.
Three
III.

Two hundred CC.
Four

IIII. or IV. Three hundred CCC.
Five
V.

Four hundred CCCC.
Six
VI.

Five hundred D. or 15.*
Seven
VII.

Six hundred DC.
Eight
VIII.

Seven hundred DCC.
Nine

VIIII. or IX. Eight hundred DCCC.
Ten
X.

Nine hundred DCCCC.
Twenty
XX.

One thousand M. or CI3.7
Thirty
XXX.

Five thousand 155. or V.
Forty

XXXX. or XL. Ten thousand CCIɔɔ. or X. Fifty L. Fifty thousand

1933. Sixty LX.

Hundred thousand CCCIDO?. or ē.
Seventy
LXX.

One million M.
Eighty
LXXX.

Two million MM. * 15 is used instead of D to represent five hundred, and for every additional 5 annexed at the right hand, the number is increased ten times.

† Cro is used to represent one thousand, and for every C and put at each end, the number is increased ten times.

* A line over any number increases its value one thousand times.

Units.

.

some

In the Arabic method the first nine numbers have each a separate character to represent it; thus, 12. A unit, or single

Note 1. These nine charthing, is represented by this

acters are called significant character,

1.

figures, because they each Two units, by this character, 2.

represent

number. Three units, by this character, 3. Sometimes, also, they are Four units, by this character, 4. called digits. Five units, by this character, 5.

Note 2. The value of Six units, by this character, 6.

these figures, as here shown,

is called their simple value. Seven units, by this character, 7.

It is their value always when Eight units, by this character, 8.

single. Nine units, by this character, 9.

Nine is the largest number which can be expressed by a single figure. There is another character, 0; it is called a cipher, naught, or nothing, because it denotes the absence of a thing. Still it is of frequent use in expressing numbers.

By these ten characters, variously combined, any number may be expressed.

The unit 1 is but a single one, and in this sense it is called a unit of the first order. All numbers expressed by one figure are units of the first order.

T3. Ten has no appropriate character to represent it, but it is considered as forming a unit of a second or higher order, consisting of tens. It is represented by the same unit figure 1 as is a single thing, but it is written at the left hand of a cipher; thus, 10, ten. The 0 fills the first place, at the right hand, which is the place of units, and the 1 the second place from the right hand, which is the place of tens. Being put in a new place, it has a new value, which is ten times its simple value, and this is what is called a local value.

Questions. 1. What is a single thing called? What is a number? Give some examples. How many ways of expressing numbers shorter than writing them out in words? What are they called? Which is the method in general use? In the Arabic method, how many nurnbers have each a separate character?

T 2. How is one represented ? Make the characters to nine. What are these nine characters called ? Why? What is the simple value of figures? What is the largest number which can be represented by a single figure? What other character is frequently used ? Why is it called naught? How many are the Arabic characters ?

What are numbers expressing single things called ?

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