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the imagination in unfolding the principle and discovering the opora tions requisite for the solution.

This principle is made the basis of this treatise ; viz. whenever a new combination is introduced, it is done with practical examples, proposed in such a manner as to show what it is, and as much as possible, how it is to be performed. The examples are so small that the pupil may easily reason upon them, and that there will be no difficulty in the operation itself, until the combination is well understood. In this way it is believed that the leading idea which the pupil will obtain of each combination, will be the effect which will be produced by it, rather than how to perform it, though the latter wilt be sufficiently well understood.

The second part contains an analytical developement of the principles. Almost all the examples used for this purpose are practical. Care has been taken to make every principle depend as little as possible upon others. Young persons cannot well follow a course of reasoning where one principle is built upon another. Besides, a principle is always less understood by every one, in proportion as it is made to depend on others.

In tracing the principles, several distinctions have been made which have not generally been made. They are principally in division of whole numbers, and in division of whole numbers by fractions, and fractions by fractions. There are some instances. also of combinations being classed together, which others have kept separate.

As the purpose is to give the learner a knowledge of the principles, it is necessary to have the variety of examples under each principle as great as possible. The usual method of arrangement, according to subjects, has been on this account entirely rejected, and the arrangeinent has been made according to principles. Many different subjects come under the same principle; and different parts of the same subject frequently come under different principles. When the principles are well understood, very few subjects will require a particular rule, and if the pupil is properly introduced to them, he will understand them better without a rule than with one. Besides, he will be better prepared for the cases which occur in business, as he will be obliged to meet them there without a name. The different subjects, as they are generally arranged, often embarrass the learner. When he meets with a name with whic, he is not

uainted, and rule attached to it, be is frequently at a loss, when if he saw the example without the name, he would not hesitate at all.

The manner of performing examples will appear new to many, but it will be found much more agreeable to the practice of men of busi

ness, and men of science generally, than those commonly found in books. This is the method of those that understand the subject. The others were invented as a substitute for understanding.

The rule of three is entirely omilted. This has been considered useless in France, for some years, though it has been retained in their books. Those who understand the principles sufficiently to comprehend the nature of the rule of three, can do much better without it than with it, for when it is used, it obscures, rather than illustrates, the subject to which it is applied. The principle of the rule of three is similar to the combinations in Art. XVI.

The rule of Position has been omitted. This is an artificial rule, the principle of which cannot be well understood without the aid of Algebra : and when Algebra is understood, Position is useless. Besides, all the examples which can be performed by Position, may be performed much more easily, and in a manner perfectly intelligible, without it. Tho manner in which they are performed is similar to that of Algebra, but without Algebraic notation. The principle of false position, proporly so called, is applied only to questions where there are not sufficient data to solve them directly.

Powers and roots, though arithmetical operations, come more properly within the province of Algebra.

There are no answers to the examples given in the book. A key is published separately for teachers, containing the answers and solutions of the most difficult examples.

XX. Contractions in division.

XXI. How to find the divisors of numbers. To find the greatest com-

mon divisor of two or more numbers. To reduce fractions to

their lowest terms.

XXII. To find the least common multiple of two or more numbers.

To reduce fractions to the least common denominator.
XXIII. To divide a whole number by a fraction, or a fraction by a

fraction, when the purpose is to find how many times the divi-
sor is contained in the dividend. To find the ratio of a fraction

and a whole number, or of two fractions.
XXIV. To divide a whole number by a fraction, or a fraction by a

fraction ; a part of a number being given to find the whole.
This is on the same principle as that of dividing a number into

parts.

XXV. Decimal Fractions. Numeration and notation of them.

XXVI. Addition and Subtraction of Decimals. To change a common

fraction to a decimal.

XXVII. Multiplication of Decimals.

XXVIII. Division of Decimals.

XXIX. Circulating Decimals.

Proof of multiplication and division by casting out Us

INDEX TO PARTICULAR SUBJECTS,

1....25

28

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Compound Multiplication

Page. Example
Addition Miscellaneous examples

37 1....49
Subtraction
Division Miscellaneous eramples

211
Interest, Simple
Commission

43....50
Insurance

92 65..113
Duties and Premiums

104 43....74
Discount, Common
Compound Interest

215 58....68

78 130..142
Discount

224 110..113

38 102..106
Barter

42 34....38

103 33....41
Loss and Gain

214 52....57

58 158..166
Fellowship, Simplo . .

85....86
Fellowship, Compound .

87....92
Equation of Payments

222 103..109
Alligation Medial .to ..

218 69....72
Alligation Alternate . .

218 73....84

79 1....49
Square and Cubic Measure. Miscellaneous Examples 91 56....64

101 13....26
Duodecimals

229 141..144
Taxes ...

103 28....32
Measure of circles, parallelograms, triangles, &c. 233 181..187
Geographical and Astronomical questions

234 188..138
Exchange

235 199..205
Tables of Coin, Weights, and Measures

236
Reflections on Mathematical reasoning

240

220
221

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