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TREATISE

ON

PLANE AND SOLID

GEOMETRY:

FOR

COLLEGES, SCHOOLS, AND PRIVATE STUDENTS.

WRITTEN FOR THE MATHEMATICAL COURSE OF

JOSEPH RAY, M. D.,

BY

ELI T. TAPPAN, M. A.,

PROFESSOR OF MATHEMATICS, MT. AUBURN INSTITUTE.

CINCINNATI:
SARGENT, WILSON & HINKLE.

CHICAGO: COBB, PRITCHARD & CO.

NEW YORK: CLARK & MAYNARD.

EducT 148.61. doa

THE BEST AND CHEAPEST.

RA Y'S

MATHEMATICAL COURSE.

Each Book complete in itself, and sold separately.

PRIMARY ARITHMETIC, or First Book : Simple Mental Lessons and

Tables. For little learners. INTELLECTUAL ARITHMETIC, OR SECOND Book: the most interesting

and valuable Intellectual Arithmetic extant. PRACTICAL ARITHMETIC, OR THIRD Book: a full and practical treat

ise on the inductive and analytic methods of instruction. For

Schools and Academies. HIGHER ARITHMETIC, or Fourth Book: the principles of Arithme

tic analyzed and practically applied. For advanced classes. TEST EXAMPLES : THREE THOUSAND practical problems for the slate

or blackboard. For drill exercises and review. Two editions-one

without Answers; the other with Answers. ELEMENTARY ALGEBRA, OR First Book : a simple, thorough, and

progressive elementary treatise. For Schools and Academies. HIGHER ALGEBRA, OR SECOND Book : a progressive, lucid, and com

prehensive work. For advanced students, and for Colleges. KEYS TO ARITHMETICS AND ALGEBRAS : embracing full and lucid

solutions to all the more difficult problems in the Intellectual, Prac

tical, and Higher Arithmetics, and the Algebras. ELEMENTS OF GEOMETRY: a comprehensive work on Plane and

Solid Geometry, with numerous practical exercises. For Colleges,

Schools, and Private Students. TRIGONOMETRY AND MENSURATION: Plane and Spherical Trigo

nometry, with their applications; Mensuration of planes and solids,

etc. (Preparing.) SURVEYING AND NAVIGATION: Surveying and Leveling, Naviga

tion, Barometric Hights, etc. (Preparing.)

To be followed by others, forming a complete Mathematical Course for Schools and Colleges.

Entered according to Act of Congress, in the year 1864, by

SARGENT, WILSON & HINKLE, In the Clerk's Office of the District Court of the United States, for the

Southern District of Ohio.

Electrotyped at the Franklin Type Foundry, Cincinnati.

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The science of Elementary Geometry, after remaining nearly stationary for two thousand years, has, for a century past, been making decided progress. This is owing, mainly, to two causes : discoveries in the higher mathematics have thrown new light upon the elements of the science; and the demands of schools, in all enlightened nations, have called out many works by able mathematicians and skillful teachers.

Professor Hayward, of Harvard University, as early as 1825, defined parallel lines as lines having the same direction. Euclid's definitions of a straight line, of an angle, and of a plane, were based on the idea of direction, which is, indeed, the essence of form. This thought, employed in all these leading definitions, adds clearness to the science and simplicity to the study. In the present work, it is sought to combine these ideas with the best methods and latest discoveries of the most distinguished writers on Geometry.

By careful arrangement of topics, the theory of each class of figures is given in uninterrupted connection. No attempt is made to exclude any method of demonstration, but rather to present examples of all. In explaining the doctrine of limits, the axiom stated by Dr. Whewell is given in the words of that eminent scholar.

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