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the imagination in unfolding the principle and discovering the operations requisite for the solution.

This principle is mado the basis of this treatise ; viz. whenever a new combination is ini roduced, it is done with practical examples, proposed in such a manner as to show what it is, and as much as pissille, 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 under gt sod. In this way it is believed that the leading idea which the pupil will obtain of each combination, will be thic effect which will be produced by it, rather than how to perform it, though the latter will be sufficiently well understood.

The second part contains an analytical developeinent of the princi. plus. 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 cunnot well follow a course of reasoning where one principle is built upon another. Besides, a prin. ciple 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 coinbinatung 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 arrangement has been made according to principles. Many different subjects come under the same principle ; and different parts of the sanie 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 un-lerstand 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 thern there without a name. The different subjects, as they are generally arranged, often embarrass the learner. When he meets with a name with which he is not acquainted, and a rule attached to it, he 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 ot' men of busi ness, and nien 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 omitted. 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 compre. hend the nature of the rule of three, can do much better without it than with it, for when it is used, it obseures, rather than illustrates, the subject to which it is applied. The principle of the aule 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. Be. sides, all the examples which can be performed by Position, may be performed much more easily, and in a manner perfectly intelligible, without it. The manner in which they are performed is similar to that of Algebra, but without Algebraic notation. The principle of false position, properly 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.

INDEX TO PARTICULAR SUBJECTS.

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103

Compound Multiplication

Page. Exaniple.
Addition Miscellaneous examples 37 1....49
Subtraction

Division Miscellaneous examples 211 1....25
Interest, Simple
Commission

28 43....5.0
Insurance

92 65..113
Duties and Premiums

104

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

215 58....68
Discount...

78 130..142
224 110..113

36 102..106
Barter

42 34....38

3:....41
Loss and Gain.....

214 52. ..57

58 138.. 160
Fellowship. Simple

220 85....86
Fellowship, Compound.....

221 87....92
Equation of Payments.

2:22 103..109
Alligation Medial..

.218 69....72
Alligation Alternate

.218 73....84

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

101 13....26
Duodecimals .....

141..144
Taxes

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

.234 188..198
Exchange.

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

.236
Reflections on Mathematical reasoning

.240

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229

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