비 유클리드 기하학. Non -Euclidean Geometry ,byHenryManning

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비 유클리드 기하학. Non -Euclidean Geometry ,byHenryManning
2000년 이전쯤의 그리스 수학자 유클리드의 기하학에서 1650년경의 뉴톤의 수학으로부터 1900년도 초에 비유클리드기하학 및 미적분학이 영국에서 발달하고 1916년도경에 아이슈타인의 상대성이론이 나옴.
이책 즉, 비유클리드기하학은 미국의 브라운대학교 교수가 1901년도에 지은책.
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.

목차

비 유클리드 기하학. 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 Formulæ . . . . . . . . . . . . . . . . . . . . . . 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

책 속으로

비 유클리드 기하학. Non -Euclidean Geometry ,byHenryManning
2000년 이전쯤의 그리스 수학자 유클리드의 기하학에서 1650년경의 뉴톤의 수학으로부터 1900년도 초에 비유클리드기하학 및 미적분학이 영국에서 발달하고 1916년도경에 아이슈타인의 상대성이론이 나옴.
이책 즉, 비유클리드기하학은 미국의 브라운대학교 교수가 1901년도에 지은책.
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.

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비 유클리드 기하학. Non -Euclidean Geometry ,byHenryManning
INTRODUCTION
The axioms of Geometry were formerly regarded as laws of thought which an
intelligent mind could neither deny nor investigate. Not only were the axioms
to which we have been accustomed found to agree with our experience, but it
was believed that we could not reason on the supposition that any of them are
not true, it has been shown, however, that it is possible to take a set of axioms,
wholly or in part contradicting those of Euclid, and build up a Geometry as
consistent as his.
We shall give the two most important Non-Euclidean Geometries.1
In these
the axioms and definitions are taken as in Euclid, with the exception of those
relating to parallel lines. Omitting the axiom on parallels,2 we are led to three
hypotheses; one of these establishes the Geometry of Euclid, while each of the
other two gives us a series of propositions both interesting and useful. Indeed, as
long as we can examine but a limited portion of the universe, it is not possible
to prove that the system of Euclid is true, rather than one of the two Non Euclidean Geometries which we are about to describe.
We shall adopt an arrangement which enables us to prove first the propositions common to the three Geometries, then to produce a series of propositions
and the trigonometrical formulæ for each of the two Geometries which differ
from that of Euclid, and by analytical methods to derive some of their most
striking properties.
We do not propose to investigate directly the foundations of Geometry, nor
even to point out all of the assumptions which have been made, consciously or
unconsciously, in this study. Leaving undisturbed that which these Geometries
have in common, we are free to fix our attention upon their differences. By a
concrete exposition it may be possible to learn more of the nature of Geometry
than from abstract theory alone.

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비 유클리드 기하학. Non -Euclidean Geometry ,byHenryManning
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