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: D. APPLETON & CO., AND PHINNEY, BLAKEMAN, & MASON.
PHILADELPHIA: J. B. LIPPINCOTT AND COMPANY.
COLUMBUS, OHIO: RILEY AND BOWLES.

PUBLIC LIBRARY
864722 A

ASTOR, LENOX AND
TILDEN FOUNDATIONS
R

1936

L

REENLEAF'S

MATHEMATICAL SERIES.

Greenleaf's New Primary Arithmetic,
Upon the Inductive Plan. Designed for Primary Schools. Improved
Electrotype Edition, beautifully illustrated. 84 pp.

Greenleaf's Intellectual Arithmetic,

Upon the Inductive Plan. Being an Advanced Intellectual Course,
for Common Schools and Academies. Improved Edition. 156 pp.

Greenleaf's Common School Arithmetic,
Or, Introduction to the National Arithmetic. A complete System
for Common Schools. Improved Electrotype Edition. 324 pp.

Greenleaf's National Arithmetic,

Being a Complete Course of Higher Arithmetic, for High Schools,
Academies, and Normal Schools. New Electrotype Edition, with
Additions and Improvements. 444 pp.

Greenleaf's Treatise on Algebra,

For Academies and High Schools, and for advanced Students in Com-
mon Schools. Improved Stereotype Edition. 360 pp.

Greenleaf's Geometry and Trigonometry,

With Practical Applications. Designed for High Schools, Academies,
and Colleges. Improved Electrotype Edition. 490 pp.

Complete Keys to the Intellectual, Common School, and National
Arithmetics, Algebra, Geometry and Trigonometry, for Teachers only.

Entered according to Act of Congress, in the year 1858, by
BENJAMIN GREENLEAF,

in the Clerk's Office of the District Court of the District of Massachusetts.

Entered according to Act of Congress, in the year 1861, by
BENJAMIN GREENLEAF,

in the Clerk's Office of the District Court of the District of Massachusetts.

Cambridge Electrotyped and Printed by Welch, Bigelow, & Co.

36 X 742

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.