OF

ALGEBRA:

INCLUDING STURMS' THEOREM.

TRANSLATED FROM THE FRENCH OF

M. BOURDON.

Gift of
James H. Russell

ADAPTED TO THE COURSE OF MATHEMATICAL INSTRUCTION IN THE
UNITED STATES,

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 ON SHADES, SHAD-

OWS, AND PERSPECTIVE.

NEW YORK:

PUBLISHED BY A. S. BARNES & CO.
No. 51 JOHN STREET.

1845.

DAVIES' ELEMENTARY GEOMETRY.

This work embraces the elementary principles of Geometry. The reasoning is plain and concise, but at the same time strictly rigorous.

DAVIES' PRACTICAL GEOMETRY,

Embracing the facts of Geometry, with applications in ARTIFICER'S WORK, MENSURATION, and MECHANICAL PHILOSOPHY.

DAVIES' BOURDON'S ALGERRA,

Being an abridgment of the work of M. Bourdon, with the addition of practical examples.

DAVIES' LEGENDRE'S GEOMETRY AND TRIGONOMETRY,

Being an abridgment of the work of M. Legendre, with the addition of a Treatise on MENSURATION OF PLANES AND SOLIDS, and a Table of LOGARITHMS and LOGARITHMIC SINES.

DAVIES' 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, and an Elementary Treatise on

NAVIGATION.

DAVIES' ANALYTICAL GEOMETRY,

Embracing the EQUATIONS OF THE POINT AND STRAIGHT LINE-of the CONIC SEC-
TIONS of the LINE AND PLANE IN SPACE-also, the discussion of the
GENERAL EQUATION of the second degree, and of SURFACES
OF THE SECOND ORDER.

DAVIES' DESCRIPTIVE GEOMETRY,

With its application to SPHERICAL PROJECTIONS.

DAVIES' SHADOWS AND LINEAR PERSPECTIVE.

DAVIES' DIFFERENTIAL AND INTEGRAL CALCULUS.

Entered, according to Act of Congress, in the year 1844,
BY CHARLES DAVIES,

in the Clerk's Office of the District Court of the United States, for the Southern District of New York.

C. A. ALVORD, PRINTER, Corner of John and Dutch Street, New York.

PREFACE

THE Treatise on Algebra, by M. Bourdon, is a work of singular excellence and merit. In France, it is one of the leading 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.

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, or to abandon it altogether.

The following work is abridged from a translation of M. Bourdon, made by Lieut. Ross, now the distinguished professor of mathematics in Kenyon College, Ohio.

The Algebra of M. Bourdon, however, has been regarded only as a standard or model. The order of arrangement, in many parts, has been changed; new rules and new methods have been introduced; and all the modifications which have