finally the analyses of theorems and problems are introduced, but in a more concrete form than usual.

The propositions are arranged with the view to obtaining a perfect logical and pedagogical order. An unusually large number of exercises is given, selected with care for the purpose of securing increased mental power.

The general plan and the preparation of the greater part of the book are the work of Dr. Schultze, while that of Dr. Sevenoak has been chiefly editorial.

PREFACE TO THE FIRST REVISED
EDITION

THE main purpose of the revision of this book has been to emphasize still further and to elaborate in greater detail the principal aim of the original edition, viz., to introduce the student systematically to original geometric work. To make the teaching of geometry both disciplinary and informational; to give to the student mental training instead of teaching him mere facts; to develop his power instead of making him memorize, these are the fundamental aims of this book.

The means employed for this purpose are similar to those used in the first edition. Still greater emphasis, however, has been placed upon the general methods which may be used for the solution of original exercises. The grading and the selection of exercises have been carefully revised. All originals that appeared unfit or too difficult have been eliminated or replaced by simpler and better ones. Topics of fundamental importance, e.g. the methods of demonstrating the equality of lines, are represented in greater detail and illustrated by a greater number of exercises than in the first edition.

In addition to these fundamental tendencies, a number of minor improvements have been introduced, among which may be mentioned:

Improved presentation of the regular propositions. Many proofs have been simplified, a more pedagogic sequence of the propositions of Book I has been adopted, Books VI and VII have been considerably simplified, and a number of difficult theorems of minor importance have been omitted or placed in the appendix.

Simplification of the so-called “incommensurable case." As this is a claim that is made by most textbooks, it may be received with some degree of skepticism, but a repeated trial of this new method will reveal its simplicity. For the more conservative teacher, however, who dislikes fundamental changes, the time-honored method is given in the appendix.

The introduction of many applied problems. These problems have been selected and arranged so as to increase the interest of the student, without sacrificing in the least the disciplinary value of the subject. Many such problems are given in the appendix.

The arrangement of the propositions and the terminology are in accord with the best modern usage. Thus, statements and reasons have been separated and placed in parallel vertical columns; the term "congruent" and the corresponding symbol are introduced and applied consistently, etc.

Many of the diagrams have been improved. The construction lines are drawn completely for most problems, graphical modes are employed for pointing out important facts, and many diagrams have been otherwise improved.

Thanks are due to Dr. J. Kahn and Mr. W. S. Schlauch for assistance in reading the proof and for helpful suggestions.

A. S.

August 1, 1913.

PREFACE TO THE PRESENT EDITION

IN the twelve years since the first revision of Schultze and Sevenoak's Geometry was made, the progress of teaching and the codifying of mathematical requirements have necessitated some changes in form and content. The present revision follows closely the first revision in organization, but has been completely rewritten with additional material. The noteworthy points are:

1. The text has been made to conform to the "Report of the National Committee on Mathematical Requirements" and to the requirements of the New York State Regents and of the College Entrance Examination Board.

2. Most of the excellent exercises of the previous edition have been kept intact and other exercises, from recent examination questions, have been added.

3. The propositions, especially in Books I and II, have been rather fully developed.

4. Additions have been made in (1) propositions, (2) simple trigonometric functions, (3) tables, (4) short review section on arithmetic and algebra, (5) a sketch of the history of geometry.

5. At the end of the various books and in the appendix, the more difficult propositions have been retained so as to meet the most exacting requirements of any college or technical school.

The authors and publishers are conscious of a great debt of gratitude to many teachers who have aided in this revision by their criticisms and suggestions. While a full list of such

collaborators is impossible, special thanks are due the following: Mr. Stephen Emery, Erasmus Hall High School, Dr. Loring B. Mullen, Girls High School, Miss Josephine D. Wilkin, Jamaica High School, Mr. Philip R. Dean, Evander Childs High School, and Mrs. Jean F. Brown, Hunter College High School, of New York City; Mr. Edmund D. Searls, High School, of New Bedford, Massachusetts; Miss A. Laura Batt, High School, of Somerville, Massachusetts; Mr. Allen H. Knapp, Central High School, and Mr. Harry B. Marsh, Technical High School, of Springfield, Massachusetts; and Miss Anna H. Andrews, Public High School, of Hartford, Connecticut.

E. S.