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This mode has also the advantage of exercising the learner in reasoning, instead of making him a listener, while the author reasons before him.

The examples in the first fifty pages involve nearly all the operations, that are ever required in simple numerical equations, with one and two unknown quantities.

In the ninth article, the learner is taught to generalize particular cases, and to form rules. Here he is first taught to represent known quantities by letters, and at the same time the purpose of it. The transition from particular cases to general principles is made as gradual as possible. At first only a part of the question is generalized, and afterwards the whole of it.

When the learner understands the purpose of representing known quantities as well as unknown, by letters or general symbols, he is considered as fairly introduced to the subject of algebra, and ready to commence where the subject is usually commenced in other treatises. Accordingly he is taught the fundamental rules, as applied to literal quantities. Much of this however is only a recapitulation in a general form, of what he has previously learnt, in a particular form.

After this, various subjects are taken up and discussed. There is nothing peculiar in the arrangement or in the manner of treating them. The author has used his own language, and explained as seemed to him best, without reference to any other work. A large number of examples introduce and illustrate every principle, and as far as seemed practicable, the subjects are taught by example rather than by explanation.

The demonstration of the Binomial Theorem is entirely original, so far as regards the rule for finding the coefficients. The rule itself is the same that has always been used. The manner of treating and demonstrating the principle of summing series by difference, is also original.*

Proportions have been discarded in algebra as well as in arithmetic. The author intended to give, in an appendix, some directions for using proportions, to assist those who might have occasion to read other treatises on mathematics. But this volume was already too large to admit it. It is believed, however, that few will find any difficulty in this respect. If they do, one hour's study of some tr a tise which explains proportions will remove it.

* See Boston Journal of Philosophy and the Arts, No. 5, for May, 1825.

In order to study this work to advantage, the learner should solve every question in course, and do it algebraically. If he finds a ques tion which he can solve as easily without the aid of algebra as with it, he may be assured, this is what the author expected. If he first solves a question, which involves no difficulty, he will understand perfectly what he is about, and he will thereby be enabled to encounter those which are difficult.

When the learner is directed to turn back and do in a new way something he has done before, let him not fail to do it, for it will be necessary to his future progress; and it will be much better to trace the new principle in what he has done before, than to have a new example for it.

The author has heard it objected to his arithmetics by some, that they are too easy. Perhaps the same objection will be made to this treatise on algebra. But in both cases, if they are too easy, it is the fault of the subject, and not of the book. For in the First Lessons, there is no explanation ; and in the Sequel there is probably less than in any other books, which explain at all. As easy however as they are, the author believes that whoever undertakes to teach them, will find the intellects of his scholars more exercised in studying them, than in studying the most difficult treatise he can put into their hands. When the learner feels, that the subject is above his capacity, he dares not attempt any thing himself, but trusts implicitly to the author ; but when he finds it level with his capacity, he readily engages in it. But here there is something more. The learner is required to perform a part himself. He finds a regular part assigned to him, and if the teacher does his duty, the learner must give a great many explanations which he does not find in the book.

TABLE OF CONTENTS.

XIII. Multiplication of compound quantities

74

XIV. Division of algebraic quantities

80

XV. Algebraic fractions

82

Multiplication of Algebraic fractions

83

XVI. Division of algebraic fractions

84

To multiply fractions by fractions

86

XVII. Reducing fractions to lower terms

89

Division when part of the factors of the divisor are con-

tained in the dividend

90

XVIII. Addition and subtraction of fractions

91

To reduce fractions to a common denominator

91

XIX. Division of whole numbers by fractions and fractions

by fractions

95

XX. Division of compound quantities

98

XXI. A few abstract examples in equations

102

XXII. Miscellaneous Questions producing simple equations 104

XXIII. Questions producing simple equations involving more

than two unknown quantities

107

XXIV. Negative quantities, explanation of them

112

XXV. Explanation of negative exponents

121

XXVI. Examination of general formulas, to see what values

the unknown quantities will take for particular suppositions

made upon the known quantities

123

XXVII. Questions producing equations of the second degree 131

XXVIII. Extraction of the second root

133

XXIX. Extraction of the second root of fractions

142

XXX. Questions producing pure equations of the second

degree

145

XXXI. Questions producing pure equations of the third de-

gree

150

Extraction of the third root

151

XXXII. Extraction of the third root of fractions

159

XXXIII. Questions producing pure equations of the third

degree

161

XXXIV. Questions producing affected equations of the

second degree

163

General formula for equations of the second degree 174

XXXV. Demonstration of the principle that every equation of

the second degree admits of two values for the unknown

quantity

175

Discussion concerning the possible and impossible val-

ues of the unknown quantity, also of the positive and

negative values of it, in equations of the second degree 177

XXXVI. Of powers and roots iz gencral

182

XXXVII. Roots of compound quantities

189

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