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BY WARREN COLBURN, A. M.
AUTHOR OF INTELLECTUAL ARITHMETIC AND SEQUEL TO DITTO.

Boston:

HILLIARD, GRAY, LITTLE, AND, WILKINS..

PUBLIC LIBRARY

ASTOR, LENOX A. TILDEN FOUNDATIONS

R 1923 L

DISTRICT OF MASSACHUSETTS, to wit.

District Clerk's Office. BE IT REMEMBERED, That on the twenty-fourth day of June, A. D. 1825, in the forty-ninth year of the Independence of the United States of America, WARREN COLBURN, of the said district, has deposited in this office the title of a book, the right whereof he claims as author, in the words following, to wit :

An Introduction to Algebra, upon the Inductive Method of Instruction. By Warren Colburn, Author of First Lessons in Arithmetic, &c."

In conformity to the act of the Congress of the United States, entitled " An act for the encouragement of learning, by securing the copies of maps, charts, and tooks, 10 the authors and proprietors of such copies, during the times therein mentioned ;” and also to an act, entitled " An act supplementary to an act, entitled An act for the encouragement of learning, by securing the copies of maps, charts, and books, to the authors and proprietors of such copies, during the times therein mentioned, and extending the benebits thereof to the arts of designing, engraving, and etching historical and other prints.”

JNO. W. DAVIS,
Clerk of the District of Massachusetts.

TABLE OF CONTENTS.

9

11

13
13

14

19

24
26

30

ntroduction. Containing a brief explanation of the purpose of

algebra, and of some of the signs

1. Questions producing simple equations, in which the unknown

quantity is multiplied by known quantities

An equation, whạt

Coefficients, what
II. Questions producing simple equations, in which the un-

known quantity is divided by known quantities
III. Questions producing simple equations, in which the un-

known quantity is added to known quantities
IV. Questions producing simple equations, in which quantities

consisting of two or more terms are to be multiplied by a
number

Transposition explained
V. Questions producing simple equations, in which quantities

consisting of two or more terms are to be divided by a num-

ber

VI. Questions producing simple equations, in which quantities

consisting of two or more terms, some of them having the

sign before them, are to be subtracted from other quanti-

ties

Case of fractions to be subtracted, when some of the terms

in the numerator have the sign
VII. Examples for exercise in putting questions into equation

A precept useful for this purpose
VIII. Questions producing equations with two unknown quan-
tities

Elimination, what

IX. Explanation of some of the higher purposes of algebra, and

examples of generalization

X. Simple quantities

Addition, multiplication, and subtraction of simple quanti-

ties

XI. Compound Quantities

Addition of compound quantities

XII. Subtraction of compound quantities

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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 forin.

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 treatise which explains proportions will remove it.

• Soe Boston Journal of Philosophy and the Arts, No. 5, for May, 1825.

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