Elements of Geometry and Trigonometry

OF

GEOMETRY AND TRIGONOMETRY.

TRANSLATED FROM THE FRENCH OF

A. M. LEGENDRE,

BY DAVID BREWSTER, LL. D.

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

BY CHARLES DAVIES,

AUTHOR OF MENTAL AND PRACTICAL ARITHMETIC, ELEMENTS OF surveying,
ELEMENTS OF DESCRIPTIVE AND OF ANALYTICAL GEOMETRY,
ELEMENTS OF DIFFERENTIAL AND INTEGRAL CALCULUS,

AND SHADES SHADOWS AND PERSPECTIVE.

PHILADELPHIA:

PUBLISHED BY A. S. BARNES AND CO.

21 Minor-street.

[1839]7

DAVIES'

COURSE OF MATHEMATICS.

DAVIES' FIRST LESSONS IN ARITHMETIC,

DESIGNED FOR BEGINNERS.

DAVIES' ARITHMETIC,

DESIGNED FOR THE USE OF ACADEMIES AND SCHOOLS.

KEY TO DAVIES' ARITHMETIC.

DAVIES' FIRST LESSONS IN ALGEBRA ;

Being an introduction to the Science, and forming a connecting link between
ARITHMETIC and ALGEBRA.

DAVIES' ELEMENTS OF 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 ALGEBRA,

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
SECTIONS 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 the Act of Congress, in the year 1834,

BY CHARLES DAVIES,

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

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THE Editor, in offering to the public Dr. Brewster's translation of Legendre's Geometry under its present form, is fully impressed with the responsibility he assumes in making alterations in a work of such deserved celebrity.

In the original work, as well as in the translations of Dr. Brewster and Professor Farrar, the propositions are not enunciated in general terms, but with reference to, and by the aid of, the particular diagrams used for the demonstrations. It is believed that this departure from the method of Euclid has been generally regretted. The propositions of Geometry are general truths, and as such, should be stated in general terms, and without reference to particular figures. The method of enunciating them by the aid of particular diagrams seems to have been adopted to avoid the difficulty which beginners experience in comprehending abstract propositions. But in avoiding this difficulty, and thus lessening, at first, the intellectual labour, the faculty of abstraction, which it is one of the primary objects of the study of Geometry to strengthen, remains, to a certain extent, unimproved.

Besides the alterations in the enunciation of the propositions, others of considerable importance have also been made in the present edition. The proposition in Book V., which proves that a polygon and circle may be made to coincide so nearly, as to differ from each other by less than any assignable quantity, has been taken from the Edinburgh Encyclopedia. It is proved in the corollaries that a polygon of an infinite number of sides becomes a circle, and this principle is made the basis of several important demonstrations in Book VIII.

Book II.,on Ratios and Proportions, has been partly adopted from the Encyclopedia Metropolitana, and will, it is believed, supply a deficiency in the original work.

Very considerable alterations have also been made in the manner of treating the subjects of Plane and Spherical Trigonometry. It has also been thought best to publish with the present edition a table of logarithms and logarithmic sines, and to apply the principles of geometry to the mensuration of surfaces and solids.

Military Academy,

West Point, March, 1834.