4416 Alexander Kiwek Euclid's Parallel Postulate: Its Nature, Validity, and Place THESIS PRESENTED TO THE PHILOSOPHICAL FACULTY 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. 1115 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. 422915 CONTENTS. I. THE PRE-LOBATCHEWSKIAN STRUGGLE WITH Historical origin of the Postulate. Attempts II. THE DISCOVERY AND DEVELOPMENT OF NON- 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- IV. PSYCHOLOGY OF THE PARALLEL POSTULATE AND General observations as to the development of I-20 |