, instead of
, as in a four-dimensional Euclidean space. Yet, because of the constancy in the values of the
’s throughout the mesh-system, hence because of the flatness or absence of non-Euclideanism or curvature of the continuum, the analogy with a Euclidean continuum is very great, and for this reason it is called semi-Euclidean.[64]
We say, therefore, that space-time is a four-dimensional semi-Euclidean continuum, and that it is differentiated from a truly Euclidean four-dimensional one solely because it has one imaginary dimension and three positive ones in place of four positive ones. In this continuum, time represents the so-called imaginary dimension, and the three dimensions of space represent the three positive ones; though it must be remembered that all we wish to imply by this statement is that there exists a difference between the dimensions of space and that of time. We should be equally justified in calling time a positive dimension, and the three dimensions of space imaginary dimensions.
Now we have also seen that the mathematical form of the expression which gives the square of the distance depends not only on the geometry of the continuum, but also on the system of co-ordinates we may have adopted. In the present case the form of the mathematical expression tells us that our co-ordinate systems or mesh-systems are necessarily Cartesian, that is, are constituted by mesh-systems of four-dimensional cubes. Hence we see that all our different Galilean mesh-systems, when taken as frames of reference, correspond to differently orientated Cartesian mesh-systems in four-dimensional space-time.[65]
In short, space-time must be regarded as the fundamental continuum of the universe. Of itself it is neither space nor time. Any Galilean observer splits it up into four directions by means of his particular Cartesian mesh-system; one of these directions corresponds to his time measurements, while the three others correspond to his space measurements.
Henceforth an instantaneous event occurring at a certain point of our Galilean system and at a definite instant of our time will be represented by one definite point in space-time, a point in four-dimensional space-time being called a Point-Event. Space-time itself appears then as a continuum of point-events, just as ordinary space was considered to be a continuum of points, and time a continuum of instants. Of course point-events, like points and instants, are mere abstractions; for a point without extension and an instant without duration are mathematical fictions introduced for purposes of definiteness; they are mere abstractions from experience.
Now just as an instantaneous event is represented by a point-event, so a prolonged event, such as a body having a prolonged existence and occupying therefore the same point or successive points in the space of our frame at successive instants of time, will be represented by a continuous line called a World-Line. If this line is straight, the body is at rest in our Galilean system, or else is animated with some constant motion along a straight line. If the world-line is curved, the motion of the body is accelerated or non-rectilinear in our Galilean system; hence it is accelerated or curvilinear in space (owing to the absolute character of acceleration). In this way all phenomena reducible to motions of particles are represented by absolute drawings in space-time.