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Alexander Kiwek

Euclid's Parallel Postulate:

Its Nature, Validity, and Place
In Geometrical Systems.

THESIS PRESENTED TO THE PHILOSOPHICAL FACULTY OF
YALE UNIVERSITY FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY.

BY

JOHN WILLIAM WITHERS, Ph.D.

President of the Teachers College, St. Louis, Mo.

CHICAGO

THE OPEN COURT PUBLISHING COMPANY

LONDON AGENTS

KEGAN PAUL, TRENCH, TRÜBNER & Co., LTD.

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PREFACE.

The parallel postulate is the only distinctive characteristic of Euclid. To pronounce upon its validity and general philosophical significance without endeavoring to know what Non-Euclideans have done would be an inexcusable blunder. For this reason I have given in the following pages what might otherwise seem to be an undue prominence to the historical aspect of my general problem.

In the last chapter, the positions taken are only briefly defended, because they seem to flow directly and naturally from results previously won.

I have included in the bibliography such works as are mentioned in the body of the thesis, and have not aimed at making a complete list. More complete biographies of Hyperspace and nonEuclidean Geometry are those of Halsted and Bonola, which I have mentioned in my list.

My obligations not elsewhere explicitly acknowledged are chiefly to Professor Geo. T. Ladd, at whose suggestion this study was undertaken, and under whose sympathetic direction it has attained its present form. I am also indebted to Dr. E. B. Wilson for light upon certain mathematical aspects of the problem.

New Haven, Connecticut, April, 1904.

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CONTENTS.

I. THE PRE-LOBATCHEWSKIAN STRUGGLE WITH

THE PARALLEL POSTULATE.

Historical origin of the Postulate. Attempts
to dispense with it. Substitution of different
definitions. Substitution of different postu-
lates. Efforts to prove the Parallel Postulate
on a basis of Euclid's other assumptions.

II. THE DISCOVERY AND DEVELOPMENT OF NON-

EUCLIDEAN SYSTEMS.

The clue. Invention of consistent geome-

tries independent of the Parallel Postulate.

The problem generalized. Attention directed

through projective methods to the qualitative

aspects of space. Recent developments

III. GENERAL ORIENTATION OF THE PROBLEM.

General significance of the historical develop-

ment of non-Euclidean geometry. The motive.

Complex character of the problem and what

its 'solution requires. Delimitation of the pres-

ent study. .

IV. PSYCHOLOGY OF THE PARALLEL POSTULATE AND

ITS KINDRED CONCEPTIONS.

General observations as to the development of
our space conception. Empirical sources of
special geometrical conceptions. Of measure-
ment and the consciousness of Homogeneity
and Free Mobility. Of the straight line. Of
Surfaces. Of the Parallel Postulate; its em-
pirical sources; its direct relations to sense-

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