Eaton's Mathematical Series.

AN

ELEMENTARY

GEOMETRY.

BY

Frottingham

WILLIAM F. BRADBURY, A. M.,

HOPKINS MASTER IN THE CAMBRIDGE HIGH SCHOOL; AUTHOR OF A TREATISE ON TRIGONOMETRY
AND SURVEYING, AND OF AN ELEMENTARY ALGEBRA.

BOSTON:

THOMPSON, BIGELOW, AND BROWN.

BRADBURY'S EATON'S ELEMENTARY ALGEBRA.

BRADBURY'S ELEMENTARY GEOMETRY.

BRADBURY'S ELEMENTARY TRIGONOMETRY.

BRADBURY'S GEOMETRY AND TRIGONOMETRY, in one volume.
BRADBURY'S TRIGONOMETRY AND SURVEYING.

KEYS OF SOLUTIONS TO COMMON SCHOOL AND HIGH
SCHOOL ARITHMETICS, TO ELEMENTARY ALGEBRA, GEOM-
ETRY, AND TRIGONOMETRY, for the use of Teachers.

Entered according to Act of Congress, in the year 1872,

BY WILLIAM F. BRADBURY,

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

UNIVERSITY PRESS: WELCH, BIGELOW, & Co.,
CAMBRIDGE.

1

PREFACE.

A LARGE number of the Theorems usually presented in textbooks of Geometry are unimportant in themselves and in no way connected with the subsequent Propositions. By spending too much time on things of little importance, the pupil is frequently unable to advance to those of the highest practical value. In this work, although no important Theorem has been omitted, not one has been introduced that is not necessary to the demonstration of the last Theorem of the five Books, namely, that in relation to the volume of a sphere. Thus the whole constitutes a single Theorem, without an unnecessary link in the chain of reasoning.

These five Books, including Ratio and Proportion, are presented in eighty-one Propositions, covering only seventy pages. This brevity has been attained by omitting all unconnected propositions, and adopting those definitions and demonstrations that lead by the shortest path to the desired end. At the close of each Book are Practical Questions, serving partly as a review, partly as practical applications of the principles of the Book, and partly as suggestions to the teacher. As those who have not had experience in discovering methods of demonstration have but little real acquaintance with Geometry, there have been added to each Book, for those who have the time and the ability, Theorems for original demonstration. These Exercises, with different methods of proving propositions already demonstrated, include those that are usually inserted, but whose demonstration in this work has been omitted. In some of these Exercises references are given to the necessary propositions; in some suggestions are made; and in a few cases the figure is constructed as the proof will require.

A sixth Book of Problems of Construction is added, which is followed by Problems for the pupil to solve. This Book, or any part of it, if thought best, can be taken immediately after completing Book III.

CAMBRIDGE, Mass., April, 1872.

W. F. B.