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Ŭ . Non -Euclidean Geometry ,byHenryManning

2000 Ŭ п 1650 κ 1900 ʿ Ŭ ߴϰ 1916濡 ̽Ÿ 뼺̷ .

å , Ŭ ̱ б 1901 å.

HENRY PARKER MANNING, Ph.D.

Assistant Professor of Pure Mathematics

in Brown University

PREFACE

Non-Euclidean Geometry is now recognized as an important branch of Mathematics. Those who teach Geometry should have some knowledge of this subject,

and all who are interested in Mathematics will find much to stimulate them and

much for them to enjoy in the novel results and views that it presents.

This book is an attempt to give a simple and direct account of the Non Euclidean Geometry, and one which presupposes but little knowledge of Mathematics. The first three chapters assume a knowledge of only Plane and Solid

Geometry and Trigonometry, and the entire book can be read by one who has

taken the mathematical courses commonly given in our colleges.

No special claim to originality can be made for what is published here. The

propositions have long been established, and in various ways. Some of the proofs

may be new, but others, as already given by writers on this subject, could not be

improved. These have come to me chiefly through the translations of Professor

George Bruce Halsted of the University of Texas.

I am particularly indebted to my friend, Arnold B. Chace, Sc.D., of Valley

Falls, R. I., with whom I have studied and discussed the subject.

HENRY P. MANNING.

Providence, January, 1901.

Ŭ . Non -Euclidean Geometry ,byHenryManning

Contents

PREFACE ii

1 INTRODUCTION 1

2 PANGEOMETRY 3

2.1 Propositions Depending Only on the Principle of Superposition . 3

2.2 Propositions Which Are True for Restricted Figures . . . . . . . 6

2.3 The Three Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . 9

3 THE HYPERBOLIC GEOMETRY 25

3.1 Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Boundary-curves and Surfaces, and Equidistant-curves and Surfaces 35

3.3 Trigonometrical Formul . . . . . . . . . . . . . . . . . . . . . . 42

4 THE ELLIPTIC GEOMETRY 51

5 ANALYTIC NON-EUCLIDEAN GEOMETRY 56

5.1 Hyperbolic Analytic Geometry . . . . . . . . . . . . . . . . . . . 56

5.2 Elliptic Analytic Geometry . . . . . . . . . . . . . . . . . . . . . 68

5.3 Elliptic Solid Analytic Geometry . . . . . . . . . . . . . . . . . . 74

6 HISTORICAL NOTE 79