Elements of Algebra

ELEMENTS OF ALGEBRA:

ON

Louis Pierre Marie

THE BASIS OF M. BOURDON:

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BY CHARLES DAVIES, LL.D.

AUTHOR OF ARITHMETIC, ELEMENTARY ALGEBRA, ELEMENTARY GEOMETRY, PRACTICAL
GEOMETRY, ELEMENTS OF SURVEYING, ELEMENTS OF DESCRIPTIVE AND

ANALYTICAL GEOMETRY, ELEMENTS OF DIFFERENTIAL

AND INTEGRAL CALCULUS, AND A TREATISE

ΟΝ SHADES, SHADOWS AND PER-

SPECTIVE.

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NEW YORK:

PUBLISHED BY A. S. BARNES & CO.,

No. 51 JOHN STREET.

1857.

1857 kw 4

Rate of Roger Brour Heldreth try of Spaceshold blop of 1843

Math 2008. DAVIES'

COURSE MATHEMATICS

Math 2058.57

HARVARD
UNIVERSITY

LIBRARY

Babies' Arithmetical Table-Book.

Davies' First Lessons in Arithmetic-For Beginners.

Davies' Arithmetic-Designed for the use of Academies and Schools,

Bey to Davies' Arithmetic.

Davies' University Arithmetic-Embracing the Science of Numbers and their numerous Applications.

Key to Davies' University Arithmetic

Davies' Elementary Algebra-Being an introduction to the Science, and forming a connecting link between ARITHMETIC and ALGEBRA.

Key to Davies Elementary Algebra.

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Davies' Elements of Geometry AND Trigonometry, with APPLICATIONS IN
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Babies' Practical Mathematics for Practical_Men-Embracing the Principles of Drawing, Architecture, Mensuration, and Logarithms, with Applications to the Mechanic Arts.

Babies' Bourdon's Algebra-Including STURM'S THEOREM-Being an abridg ment of the Work of M. BOURDON, with the addition of practical examples.

Davies' Legendre's Geometry AND Trigonometry-From the works of A. M.
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Dabies' Surveying-With a description and plates of the THEODOLITE, COMPASS, PLANE-TABLE, and LEVEL; also, Maps of the TOPOGRAPHICAL SIGNS adopted by the Engineer Department—an explanation of the method of surveying the Public Lands, Geodesic and Maritime Surveying, and an Elementary Treatise on NAVIGATION.

Davies' Descriptive Geometry-With its application to SPHERICAL PROJEC

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Davies' Shades, Shadows, and Linear Perspective.

Davies' Analytical Geometry-Embracing the EQUATIONS OF THE Point and STRAIGHT LINE-of the CONIC SECTIONS-of the LINE AND PLANE IN SPACE; also, the discussion of the GENERAL EQUATION of the second degree, and of SUB

FACES of the second order.

Babies' Differential and Entegral Calculus.

Dabies' Logic and Utility of Mathematics.

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ENTERED according to Act of Congress, in the year one thousand eight hundred and

fifty-three by CHARLES DAVIES, in the Clerk's Office of the District Court of the United States for the Southern District of New York.

J P. JONES & CO., STEREOTYPERS.

PREFACE.

THE Treatise on Algebra, by M. Bourdon, is a work of singular excellence and merit. In France, it has long been one of the standard Text books. Shortly after its first publication, it passed through several editions, and has formed the basis of every subsequent work on the subject of Algebra, both in Europe and in this country.

The original work is, however, a full and complete treatise on the subject of Algebra, the later editions containing about eight hundred pages octavo. The time which is given to the study of Algebra, in this country, even in those seminaries where the course of mathematics is the fullest, is too short to accomplish so voluminous a work, and hence it has been found necessary either to modify it essentially, or to abandon it altogether.

In the following work, the original Treatise of Bourdon has been regarded only as a model. The order of arrangement, in many parts, has been changed; new rules and new methods have been introduced: the modifications indicated by its use, for twenty years, as a text book

in the Military Academy have been freely made, for the purpose of giving to the work a more practical character, and bringing it into closer harmony with the trains of thought and improved systems of instruction which prevail in that institution.

But the work, in its present form, is greatly indebted to the labors of William G. Peck, A. M., U. S. Topographical Engineers, and Assistant Professor of Mathematics in the Military Academy.

Many of the new definitions, new rules and improved methods of illustration, are his. His experience as a teacher of mathematics has enabled him to bestow upon the work much valuable labor which will be found to bear the marks of profound study and the freshness of daily instruction.

FISHKILL LANDING,

CONTENTS.

CHAPTER I.

DEFINITIONS AND PRELIMINARY REMARKS.

ALGEBRA Definitions-Explanation of the Algebraic Signs.

Theorems-Problems-Definition of-Problem..