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.
But this point of view by no means embraces the whole of the actual
process; for it slurs over the important part played by intuition and
deductive thought in the development of an exact science. As soon as a
science has emerged from its initial stages, theoretical advances are
no longer achieved merely by a process of arrangement. Guided by
empirical data, the investigator rather develops a system of thought
which, in general, is built up logically from a small number of
fundamental assumptions, the so-called axioms. We call such a system of
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