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OF

GEOMETRY AND TRIGONOMETRY;

WITH

PRACTICAL APPLICATIONS.

BY BENJAMIN GREENLEAF, A. M.,

AUTHOR OF A MATHEMATICAL SERIES.

IMPROVED ELECTROTYPE EDITION.

BOSTON:
PUBLISHED BY ROBERT S. DAVIS & CO.

NEW YORK: OAKLEY & MASON, AND A. S. BARNES & co.

PHILADELPHIA: J. A. BANCROFT & COMPANY.

ST. LOUIS: KEITH AND WOODS.

CHICAGO: 8. C. GRIGGS & CO.

GREENLEAF'S

NEW COMPREHENSIVE SERIES.

AN ENTIRELY NEW MATHEMATICAL COURSE, fully adapted to

the best methods of Modern Instruction.

GREENLEAF'S NEW PRIMARY ARITHMETIC.

GREENLEAF'S NEW INTELLECTUAL ARITHMETIC.

GREENLEAF'S NEW ELEMENTARY ARITHMETIC.

GREENLEAF'S NEW PRACTICAL ARITHMETIC.

GREENLEAF'S NEW ELEMENTARY ALGEBRA.

GREENLEAF'S NEW HIGHER ALGEBRA.

GREENLEAF'S ELEMENTS OF GEOMETRY.

GREENLEAF'S ELEMENTS OF TRIGONOMETRY.

GREENLEAF'S GEOMETRY AND TRIGONOMETRY.

e Other Books of a Complete Series, in preparation.

KEYS to the PRACTICAL ARITHMETIC, ALGEBRAS, GEOMETRY and TRIGONOMETRY, in

separate volumes.

Entered according to Act of Congress, in the year 1863, by

BENJAMIN GREENLEAF,
in the Clerk's Office of the District Court for the District of Massachusetts

RIVERSIDE PRESS :

PRINTED BY .. 0. HOUGITON AND COMPANY.

HARVARD
UNIVERSITY
LIBRARY

PREFACE.

The preparation of this treatise was undertaken at the earnest solicitation of many teachers, who, having used the author's Arithmetics and Algebra with satisfaction, were desirous of seeing his series rendered more nearly complete by the addition of the Elements of Geometry and Trigonometry.

That there are peculiar advantages in a graded series of textbooks on the same subject, few, if any, properly qualified to judge, will doubt. The author, therefore, feels justified in introducing this volume to the attention of the public.

In the Elements of Geometry, he has followed, in the main, the simple and elegant order of arrangement adopted by Legendre; but in the methods of demonstration no particular authority has been closely followed, the aim having been to adapt the work fully to the latest and most approved modes of instruction. In this respect, there will be found incorporated a considerable number of important improvements.

More attention than is usual in elementary works of this kind has been given to the converse of propositions. In almost all cases where it was possible, the converse of a proposition has been demonstrated.

The demonstration of Proposition XX. of the first book is essentially the one given by M. da Cunha in the Principes Mathématiques, which has justly been pronounced by the highest mathematical authorities to be a very important improvement in elementary geometry. It has, however, never before been introduced into a text-book by an American author.

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