Here, however, the reader may be reminded that even for those who are interested solely in trends of thought or in the evolution of ideas, no popular or semi-popular book can ever aspire to take the place of the highly technical mathematical works. The superiority of the latter lies not in the bare mathematical formulæ which they contain. Rather does it reside in the power the mathematical instrument has of giving us a deeper insight into the problems of nature, revealing unsuspected harmonies and extending our survey into regions of thought whence the human intelligence would otherwise be excluded. Thus the sole rôle a semi-popular book can hope to perform is to serve as a general introduction, to whet the appetite for further knowledge, if a craving for knowledge is within us. To presume, as the philosophers do, that a vague understanding of a highly technical subject, gleaned from semi-popular writings, or from the snatching at a sentence here and there in a technical book, should enable them to expound a theory, criticise it, and, worse still, ornament it with their own ideas, is an opinion which has done much to create a spirit of distrust towards their writings. The answer Euclid gave to King Ptolemy, “There is no royal road, no short cut to knowledge,” remains true to-day, still [Pg vi] truer than in the days of ancient Alexandria, when science had not yet grown to the proportions of a mighty tree. I wish to take this opportunity to express my gratitude to Prof. Leigh Page of Yale University for his kindness in looking over the manuscript and offering many valuable suggestions. A. D’ Abro. NEW YORK, 1927. [Pg vii] CONTENTS PAGE FOREWORD ix PART I PRE-RELATIVITY PHYSICS CHAPTER I. MANIFOLDS 23 II. THE BIRTH OF METRICAL GEOMETRY 32 III. RIEMANN’S DISCOVERIES AND CONGRUENCE 39 IV. THE PROBLEM OF PHYSICAL SPACE 47 V. AN ALTERNATIVE VIEW OF NON-EUCLIDEAN GEOMETRIES 60 VI. TIME 71 VII. SYSTEMS OF CO-ORDINATES AND DISTANCE 83 VIII. THE MEANING OF THE WORD RELATIVITY 99 IX. THE PRINCIPLES OF RELATIVITY 103 X. CLASSICAL MECHANICS AND THE NEWTONIAN PRINCIPLE OF RELATIVITY 106 XI. THE ETHER 116 XII. THE EQUATIONS OF ELECTROMAGNETICS AND LORENTZ’S THEORY 125 PART II THE SPECIAL THEORY OF RELATIVITY XIII. EINSTEIN’S SPECIAL THEORY OF RELATIVITY 143 XIV. RELATIVISTIC MECHANICS 156 XV. CONSEQUENCES OF THE NEW SPACE AND TIME MEASUREMENTS—SIMULTANEITY 161 XVI. PRACTICAL CONGRUENCE IN RELATIVITY 187 XVII. THE MATHEMATICAL EXPRESSION OF EINSTEIN’S FUNDAMENTAL PREMISES 193 XVIII. THE DISCOVERY OF SPACE-TIME 195 XIX. VARIOUS POSSIBLE WORLDS 201 XX. THE IRREVERSIBILITY OF TIME 212 XXI. THE REALITY OF THE CONTRACTION OF LENGTHS AND OF THE LENGTHENING OF DURATIONS 219 XXII. THE PARADOXES ASSOCIATED WITH SPACE-TIME AND THE TRIP TO THE STAR 225 [Pg viii] PART III THE GENERAL THEORY OF RELATIVITY XXIII. POTENTIALS AND FORCES