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From the 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 non-Euclidean 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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