INTRODUCTION: Explanatory Notes x _PART ONE_ CHAPTER I THE PROLOGUE On the Variability of Psychic Powers--The Discovery of the Fourth Dimension Marks a Distinct Stage in Psychogenesis--The non-Methodical Character of Discoveries--The Three Periods of Psychogenetic Development--The Scope and Permissibility of Mathetic License--Kosmic Unitariness Underlying Diversity 23 CHAPTER II HISTORICAL SKETCH OF THE HYPERSPACE MOVEMENT Egypt the Birthplace of Geometry--Precursors: Nasir-Eddin, Christoph Clavius, Saccheri, Lambert, La Grange, Kant--Influence of the _Mecanique Analytique_--The Parallel Postulate the Root and Substance of the Non-Euclidean Geometry--The Three Great Periods: The Formative, Determinative and Elaborative--Riemann and the Properties of Analytic Spaces 44 CHAPTER III ESSENTIALS OF THE NON-EUCLIDEAN GEOMETRY The Non-Euclidean Geometry Concerned with Conceptual Space Entirely--Outcome of Failures at Solving the Parallel-Postulate--The Basis of the Non-Euclidean Geometry--Space Curvature and Manifoldness--Some Elements of the Non-Euclidean Geometry--Certainty, Necessity and Universality as Bulwarks of Geometry--Some Consequences of Efforts at Solving the Parallel-Postulate--The Final Issue of the Non-Euclidean Geometry--Extended Consciousness 69 CHAPTER IV DIMENSIONALITY Arbitrary Character of Dimensionality--Various Definitions of Dimension--Real Space and Geometric Space Differentiated--The Finity of Space--Difference Between the Purely Formal and the Actual--Space as Dynamic Appearance--The _A Priori_ and the _A Posteriori_ as Defined by Paul Carus 92 CHAPTER V THE FOURTH DIMENSION The Ideal and the Representative Nature of Objects in the Sensible World--The Fluxional, the Basis of Mental Differences--Natural Symbols and Artificial Symbols--Use of Analogies to Prove the Existence of a Fourth Dimension--The Generation of a Hypercube or Tesseract--Possibilities in the World of the Fourth Dimension--Some Logical Difficulties Inhering in the Four-Space Conception--The Fallacy of the Plane-Rotation Hypothesis--C. H. Hinton and Major Ellis on the Fourth Dimension 118 _PART TWO_ SPATIALITY: AN INQUIRY INTO THE ESSENTIAL NATURE OF SPACE AS DISTINGUISHED FROM THE MATHEMATICAL INTERPRETATION CHAPTER VI CONSCIOUSNESS THE NORM OF SPACE DETERMINATIONS