APPENDIX II MINKOWSKI’S FOUR-DIMENSIONAL SPACE (“WORLD”) (SUPPLEMENTARY TO SECTION XVII) We can characterise the Lorentz transformation still more simply if we introduce the imaginary image031 in place of _t_, as time-variable. If, in accordance with this, we insert image050 and similarly for the accented system _K′_, then the condition which is identically satisfied by the transformation can be expressed thus: _x_1′2 + _x_2′2 + _x_3′2 + _x_4′2 = _x_12 + _x_22 + _x_32 + _x_42 (12). That is, by the afore-mentioned choice of “coordinates,” (11_a_) [see the end of Appendix II] is transformed into this equation. We see from (12) that the imaginary time co-ordinate _x_4, enters into the condition of transformation in exactly the same way as the space co-ordinates _x_1, _x_2, _x_3. It is due to this fact that, according to the theory of relativity, the “time” _x_4, enters into natural laws in the same form as the space co ordinates _x_1, _x_2, _x_3. A four-dimensional continuum described by the “co-ordinates” _x_1, _x_2, _x_3, _x_4, was called “world” by Minkowski, who also termed a point-event a “world-point.” From a “happening” in three-dimensional space, physics becomes, as it were, an “existence” in the four-dimensional “world.” This four-dimensional “world” bears a close similarity to the three-dimensional “space” of (Euclidean) analytical geometry. If we introduce into the latter a new Cartesian co-ordinate system (_x′_1, _x′_2, _x′_3) with the same origin, then _x′_1, _x′_2, _x′_3, are linear homogeneous functions of _x_1, _x_2, _x_3 which identically satisfy the equation _x_1′2 + _x_2′2 + _x_3′2 = _x_12 + _x_22 + _x_32 The analogy with (12) is a complete one. We can regard Minkowski’s “world” in a formal manner as a four-dimensional Euclidean space (with an imaginary time coordinate); the Lorentz transformation corresponds to a “rotation” of the co-ordinate system in the four-dimensional “world.” APPENDIX III THE EXPERIMENTAL CONFIRMATION OF THE GENERAL THEORY OF RELATIVITY From a systematic theoretical point of view, we may imagine the process of evolution of an empirical science to be a continuous process of induction. Theories are evolved and are expressed in short compass as statements of a large number of individual observations in the form of empirical laws, from which the general laws can be ascertained by comparison. Regarded in this way, the development of a science bears some resemblance to the compilation of a classified catalogue. It is, as it were, a purely empirical enterprise.