My thanks are due to the friends who have assisted me in designing and preparing the modifications of my previous models, and in no small degree to the publisher of this volume, Mr. Sonnenschein, to whose unique appreciation of the line of thought of this, as of my former essays, their publication is owing. By the provision of a coloured plate, in addition to the other illustrations, he has added greatly to the convenience of the reader. C. HOWARD HINTON. CONTENTS CHAP. PAGE I. FOUR-DIMENSIONAL SPACE 1 II. THE ANALOGY OF A PLANE WORLD 6 III. THE SIGNIFICANCE OF A FOUR-DIMENSIONAL EXISTENCE 15 IV. THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE 23 V. THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 41 Lobatchewsky, Bolyai, and Gauss Metageometry VI. THE HIGHER WORLD 61 VII. THE EVIDENCE FOR A FOURTH DIMENSION 76 VIII. THE USE OF FOUR DIMENSIONS IN THOUGHT 85 IX. APPLICATION TO KANT’S THEORY OF EXPERIENCE 107 X. A FOUR-DIMENSIONAL FIGURE 122 XI. NOMENCLATURE AND ANALOGIES 136 XII. THE SIMPLEST FOUR-DIMENSIONAL SOLID 157 XIII. REMARKS ON THE FIGURES 178 XIV. A RECAPITULATION AND EXTENSION OF THE PHYSICAL ARGUMENT 203 APPENDIX I.—THE MODELS 231 " II.—A LANGUAGE OF SPACE 248 THE FOURTH DIMENSION CHAPTER I FOUR-DIMENSIONAL SPACE There is nothing more indefinite, and at the same time more real, than that which we indicate when we speak of the “higher.” In our social life we see it evidenced in a greater complexity of relations. But this complexity is not all. There is, at the same time, a contact with, an apprehension of, something more fundamental, more real.