BY ELIAS LOOMIS, LL.D.,

PROFESSOR OF NATURAL PHILOSOPHY AND ASTRONOMY IN YALE COLLEGE,
AND AUTHOR OF "A COURSE OF MATHEMATICS."

REVISED EDITION, WITH APPENDIX.

NEW YORK:

HARPER & BROTHERS, PUBLISHERS,

FRANKLIN SQUARE.

HARVARD
UNIVERSITY
LIBRARY
0477172

LOOMIS'S SERIES OF TEXT-BOOKS.

pp., 90 cents.
128 pp., 90 cents.
pp., $1 00.

ELEMENTARY ARITHMETIC. 166 pp., 28 cents.
TREATISE ON ARITHMETIC. 352 pp., 88 cents.
ELEMENTS OF ALGEBRA. Revised Edition. 281
Key to Elements of Algebra, for Use of Teachers.
TREATISE ON ALGEBRA. Revised Edition. 384
Key to Treatise on Algebra, for Use of Teachers.
ALGEBRAIC PROBLEMS AND EXAMPLES. 258 pp., 90 cents.
ELEMENTS OF GEOMETRY. Revised Edition. 388 pp., $1 00.

219 pp., $1 00.

ELEMENTS OF TRIGONOMETRY, SURVEYING, AND NAVIGATION. 194 pp., $1 00. TABLES OF LOGARITHMS. 150 pp., $1 00.

The Trigonometry and Tables, bound in one volume. 360 pp., $1 50.

ELEMENTS OF ANALYTICAL GEOMETRY. Revised Edition. 261 pp., $1 00.
DIFFERENTIAL AND INTEGRAL CALCULUS. Revised Edition. 309 pp., $1 00.
The Analytical Geometry and Calculus, bound in one volume. 570 pp., $1 75.
ELEMENTS OF NATURAL PHILOSOPHY. 351 pp., $1 05.

ELEMENTS OF ASTRONOMY. 254 pp., $1 00.

PRACTICAL ASTRONOMY. 499 pp., $1 50.
TREATISE ON ASTRONOMY. 351 pp., $1 50.
TREATISE ON METEOROLOGY. 308 pp., $1 50.

Entered according to Act of Congress, in the year 1876, by
HARPER & BROTHERS,

In the Office of the Librarian of Congress, at Washington.

PREFAСЕ.

THE stereotype plates of my Elements of Geometry and Conic Sections having become so much worn by long-continued use that it was found necessary to recast them, the opportunity has been improved to give the entire book a thorough revision. As the general plan of the original work has met with very extensive approval, it has not been thought best to modify it materially; nevertheless, the minor changes which have been made are numerous and of considerable importance.

The volume commences with a brief sketch of the history of Elementary Geometry, which, it is hoped, may increase the student's interest in a subject which has occupied the attention of so many gifted minds. The definitions of Book I. have been somewhat amplified, for the purpose of giving clearer ideas of the philosophy of the subject; and several notes have been added to the first pages of the book which, it is hoped, may be found useful and suggestive, although they are generally such as any competent teacher might easily have supplied.

In Book II. the subject of Ratio has been expanded, especially for the purpose of meeting the difficulty of incommensurable quantities; and in this I have followed substantially the method of Vincent in his Cours de Géométrie. A few new propositions have been added to Books III., IV., and V.; and at the close of Book VI. is given a considerable collection of new theorems and problems, with some numerical exercises on the preceding books. These theorems and problems are so simple that it is hoped many students may be encouraged to labor upon them; for no one can be considered as master of the subject of Geometry who has not acquired the ability to discover the demonstration of new theorems and the solution of new problems. Those who find these