PREFACE

IT has been the aim of the author to make this a teachable book. Critical attention has been given to the minutest details. In the earlier pages most corollaries are proved, and most references, including the postulates, are quoted in full. The subject-matter is so arranged that no page must be turned in reading a demonstration.

Brevity. - While brevity is not made an end in itself, yet brevity is an element of clearness. The object at all times has been to present the subject in as clear, simple, and direct a manner as possible. The author's long experience in the class-room has taught him that the fewer the words in which a thing can be said the better; elaborate explanations beget confusion.

Propositions Omitted. — In the Plane Geometry only those theorems forming links in the chain of logic leading up to the mensuration of the circle, and a few required in the treatment of Solid Geometry, have been admitted. Familiar propositions omitted will be found among the exercises. No theorem, however, has been omitted which is necessary to meet the entrance requirements of colleges and technical schools.

Definitions. Definitions may be defective in many ways, but an essential feature of any definition is that its converse must be true. The reason for this is that we frequently assume the truth of such converses in our reasoning.

Great pains have been taken to make the definitions clear, concise, and correct. No person can reason correctly on any subject without a clear conception of the meaning of the terms used. If the student is to be broadly grounded in an exact science, he must cognize the exact meaning of the terms peculiar to that science. The point, line, surface, and solid are the fundamental concepts in Geometry. A clear mental grasp of these concepts is all-important to the pupil's progress. Hence, in §§ 27 to 44 these concepts are developed, illustrated, and explained. Then follow their mathematical definitions.

Exercises. — The book contains a large number of exercises for original work on the part of the student. They are well proportioned between theorems, problems, problems of computation, and problems in loci. These exercises have been collected with great care and labor, extending over many years of patient research. Many are original with the author. The others have been gleaned from a common field of wide range. The aim has been to give only exercises which have a direct educational value, or which give a distinct property of some geometric figure. Curios and puzzles have been scrupulously avoided. All these exercises have been carefully graded. Those within the body of each book are, for obvious reasons, based upon the proposition, or propositions, which they immediately follow. Those at the end of each book are, in general, more difficult. It is not expected that any one class will find time to work all these exercises. The judgment of the teacher, guided by local conditions, must be depended upon to make suitable selections. The great value of original work on the part of the student is now universally admitted.

It is the aim of this book to lead the student into this original work so gradually, so systematically, that in a short time he will grow enthusiastic over "originals." Later on full directions are given for attacking original theorems and for analyzing problems.

Special attention is called to the large number of numerical exercises. The author believes that numerical exercises are the best means at our command for impressing geometric truths upon the mind of the student. These exercises are not given for the purpose of teaching arithmetic. Hence they are exceedingly simple, and can be solved rapidly. A simple numerical problem illustrates a principle better than a complex one. Moreover, the student in Geometry has no time for complicated mental gymnastics.

Limits. Every principle in limits is given which is required to meet the demands of logic at every point in the Plane and Solid Geometry. The author believes that the subject as here presented is clearly within the mental grasp of the average student. He has found it so in his own classes.

Loci. The subject of loci is a very simple one when properly presented. It is of much greater importance than usually supposed. In the analysis of many problems, its use is invaluable. The author has aimed to give the subject a treatment commensurate with its importance. reasons, the subject is introduced under circles.

For obvious

Maxima and Minima, also Symmetry, are added as supplementary. Students who take Plane Geometry only may omit these subjects; those who take Solid Geometry may omit Maxima and Minima, but should take Symmetry.

The author takes great pleasure in acknowledging his indebtedness to Dr. Charles Robert Gaston, and to Professor Leland L. Landers, of the Richmond Hill High School, for valuable suggestions, and for their kindness in reading the proof-sheets.

GEOMETRY

INTRODUCTION

GENERAL TERMS

1 A proposition is the expression of a judgment.

2 Propositions are distinguished as definitions, theorems, problems, corollaries, lemmas, axioms, postulates, absurdities, and scholiums.

3 A definition is such a description of an object as serves to distinguish it from all other objects.

The converse of a correct definition is always true.

4 A theorem is a truth which may be logically demonstrated. 5 A problem is a question proposed for solution.

6 A corollary is a truth easily deduced from one or more propositions.

7 A lemma is an auxiliary theorem or problem.

8 An axiom is a self-evident truth.

9 A postulate is a self-evident possibility.

10 An absurdity is a self-evident falsity.

11 A scholium is a remark made upon one or more propositions.

12 A demonstration is the proof of a theorem.

13 A solution is the process of solving a problem.

14 A demonstration is direct or indirect.