Chapter I (part 1)

p. vii CONTENTS AUTHOR'S PREFACE TO THE SECOND EDITION xiii INTRODUCTION TO THE ENGLISH TRANSLATION 1 CHAPTER I What do we know and what do we not know? Our data, and the things for which we seek. The unknown mistaken for the known. Matter and motion. What does the positive philosophy come to? Identity of the unknown: x=y, y=x. What we really know. The existence of consciousness in us, and of the world outside of us. Dualism or monism.? Subjective and objective knowledge. Where do the causes of the sensations lie? Kant's system. Time and Space. Kant and the "ether." Mach's observation. With what does the physicist really deal? 11 CHAPTER II A new view of the Kantian problem. The books of Hinton. The "space-sense" and its evolution. A system for the development of a sense of the fourth dimension by exercises with colored cubes. The geometrical conception of space. Three perpendiculars—why three? Can everything existing be measured by three perpendiculars? The indices of existence. Reality of ideas. Insufficient evidence of the existence of matter and motion. Matter and motion are only logical concepts, like "good" and "evil." 23 CHAPTER III What may we learn about the fourth dimension by a study of the geometrical relations within our space? What should be the relation between a three-dimensional body and one of four dimensions? The four-dimensional body as the tracing of the movement of a three-dimensional body in the direction which is not confined within it. A four-dimensional body as containing an infinite number of three-dimensional bodies. A three dimensional body as a section of a four-dimensional one. Parts of bodies and entire bodies in three and in four dimensions. The incommensurability of a three-dimensional and a four-dimensional body. A material atom as a section of a four-dimensional line. 34 CHAPTER IV In what direction may the fourth dimension lie? What is motion? Two kinds of motion—motion in space and motion in time—which are contained in every movement. What is time? Two ideas contained in the conception of time. The new dimension of space, and motion upon that dimension. Time as the fourth dimension of space. Impossibility of understanding the fourth dimension without the idea of motion. The idea of motion and the "time-sense." The time sense as a limit (surface) of the "space-sense." Hinton on the law of surfaces. p. viii The "ether" as a surface. Riemann's idea concerning the translation of time into space in the fourth dimension. Present, past, and future. Why do we not see the past and the future. Life as a feeling of one's way . Wundt on the subject of our sensuous knowledge. 33 CHAPTER V Four-dimensional space. "Temporal body"— Linga Sharîra . The form of a human body from birth to death. Incommensurability of three-dimensional and four-dimensional bodies. Newton's fluents. The unreality of constant quantities in our world. The right and left hands in three-dimensional and in four dimensional space. Difference between three-dimensional and four-dimensional space. Not two different spaces but different methods of receptivity of one and the same world 52 CHAPTER VI Methods of investigation of the problem of higher dimensions. The analogy between imaginary worlds of different dimensions. The one-dimensional world on a line. "Space" and "time" of a one-dimensional being. The two-dimensional world on a plane. "Space" and "time," "ether," "matter," and "motion" of a two-dimensional being. Reality and illusion on a plane. The impossibility of seeing an "angle." An angle as motion. The incomprehensibility to a two-dimensional being of the functions of things in our world.