OF

GEOMETRY AND TRIGONOMETRY,

FROM THE WORKS OF

A. M. LEGENDRE.

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

BY CHARLES DAVIES, LL.D.,

AUTHOR OF ARITHMETIC, ALGEBRA, PRACTICAL MATHEMATICS FOR PRACTICAL MEN,
ELEMENTS OF DESCRIPTIVE AND OF ANALYTICAL GEOMETRY, ELEMENTS

OF DIFFERENTIAL AND INTEGRAL CALCULUS, AND SHADES,
SHADOWS, AND PERSPECTIVE.

A. S. BARNES & COMPANY,

NEW YORK AND CHICAGO.

1869.

FluCT $48.69.516

HARVARD COL' TGE LIDRARY

GIFT OF

GRENVILLE H. NORCROSS

1937

Davies' Course in Mathematics.

THE OLDEST AND MOST THOROUGH OF ALL MATHEMATICAL SERIES:
LATELY REVISED.

Babies' New Primary Arithmetic-Designed for Beginners; containing the
elementary tables of Addition, Subtraction, Multiplication, Division, and
Denominate Numbers; with a large number of easy and practical ques-
tions, both mental and written.

Davies' First Lessons in Arithmetic-Combining the Oral Method, with
the Method of Teaching the Combinations of Figures by Sight.
Davies' New Entellectual Arithmetic-An Analysis of the Science of Num-
bers, with especial reference to Mental Training and Development.
Davies' New Practical Arithmetic and Bey-Combines the best Features
of the New School Arithmetic, with important additions. It is Educa-
tional and Practical.

Davies' Elements of Written Arithmetic-A most Popular and Complete
Text-book. More advanced than the Primary, and an introduction to the
New Practical Arithmetic. Teaches the written language of Arithmetic.
Dabies' Grammar of Arithmetíc—An Analysis of the Language of Num-
bers and the Science of Figures.

Davies' New University Arithmetic and Bey-Embracing the Science of
Numbers, and their Applications according to the most Improved Meth-
ods of Analysis and Cancellation.

Davies' New Elementary Algevra and Key—Embracing the First Princi-
ples of the Science.

Davies' Outlines of Mathematical Science-A Concise Explication of the
Science. Valuable for Study and Reference.

Babies' Elementary Geometry AND Trigonometry-With Applications in
Mensuration.

Davies' Practical Mathematics-With Drawing and Mensuration applied to
the Mechanic Arts.

Davies' University Algebra-Embracing a Logical Development of the
Science, with graded examples.

Babies' Bourdon's Algebra-Including Sturm's and Horner's Theorems,
and practical examples.

Babies' Legendre's Geometry and Trigonometry-Revised and adapted to the course of Mathematical Instruction in the United States.

Babies' Elements of Surveying AND Navigation-Containing descriptions
of the Instruments and necessary Tables.

Davies' Analytical Geometry-Embracing the Equations of the Point, the
Straight Line, the Conic Sections, and Surfaces of the first and second order.
Babies' Differential AND Entegral Calculus.

Babies' Descriptive Geometry-With its application to Spherical Trigonome-
try, Spherical Projections, and Warped Surfaces.

Babies' Shades, Shadows, and Perspective.

Babies' Logic and Utility of Mathematics-With the best methods of Instruction Explained and Illustrated.

Babies' and Peck's Mathematical Dictionary and Cyclopedia of Mathematical Science-Comprising Definitions of all the terms employed in Mathematics-an Analysis of each Branch, and of the whole, as forming a single Science.

ENTERED, according to Act of Congress, in the year one thousand eight hundred and sixty-two, by CHARLES DAVIES, in the Clerk's Office of the District Court of the United States for the Southern District of New York.

PREFACE.

Or the various Treatises on Elementary Geometry which have appeared during the present century, that of M. LEGENDRE stands preeminent. Its peculiar merits have won for it not only a European reputation, but have also caused it to be selected as the basis of many of the best works on the subject that have been published in this country.

In the original Treatise of LEGENDRE, the propositions: are not enunciated in general terms, but by means of the diagrams employed in their demonstration. This departure from the method of EUCLID is much to be regretted. The propositions of Geometry are general truths, and ought to be stated in general terms, without reference to particular diagrams. In the following work, each proposition is first enunciated in general terms, and afterwards, with reference to a particular figure, that figure being taken to represent any one of the class to which it belongs. By this arrangement, the difficulty experienced by beginners in comprehending abstract truths, is lessened, without in any manner impairing the generality of the truths evolved.

The term solid, used not only by LEGENDRE, but by many other authors, to denote a limited portion of space, seems calculated to introduce the foreign idea of matter