CHAPTER I.--On the Nature of Mathematical Reasoning 31 Syllogistic Deduction 31 Verification and Proof 32 Elements of Arithmetic 33 Reasoning by Recurrence 37 Induction 40 Mathematical Construction 41 CHAPTER II.--Mathematical Magnitude and Experience 43 Definition of Incommensurables 44 The Physical Continuum 46 Creation of the Mathematical Continuum 46 Measurable Magnitude 49 Various Remarks (Curves without Tangents) 50 The Physical Continuum of Several Dimensions 52 The Mathematical Continuum of Several Dimensions 53 PART II. _Space_ CHAPTER III.--The Non-Euclidean Geometries 55 The Bolyai-Lobachevski Geometry 56 Riemann's Geometry 57 The Surfaces of Constant Curvature 58 Interpretation of Non-Euclidean Geometries 59 The Implicit Axioms 60 The Fourth Geometry 62 Lie's Theorem 62 Riemann's Geometries 63 On the Nature of Axioms 63 CHAPTER IV.--Space and Geometry 66 Geometric Space and Perceptual Space 66 Visual Space 67 Tactile Space and Motor Space 68 Characteristics of Perceptual Space 69 Change of State and Change of Position 70 Conditions of Compensation 72 Solid Bodies and Geometry 72 Law of Homogeneity 74 The Non-Euclidean World 75 The World of Four Dimensions 78 Conclusions 79 CHAPTER V.--Experience and Geometry 81 Geometry and Astronomy 81 The Law of Relativity 83 Bearing of Experiments 86 Supplement (What is a Point?) 89 Ancestral Experience 91 PART III. _Force_