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Euclid's Parallel Postulaze:
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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4-24-1923

COPYRIGHT 1905

BY
THE OPEN COURT PUBLISHING CO.

CHICAGO

PREFACE.

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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 non-
Euclidean Geometry are those of Halsted and Bon-
ola, which I have mentioned in my list.

My obligations not elsewhere explicitly acknowl-
edged 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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