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 Here, it would not be hard to introduce the reduced velocity v, as well as to calculate the deviation of the resultant in the general case, where the components enclose an arbitrary angle with each other.

If we compose three to each other perpendicular velocities in space, then we arrive at six resultants of different direction.

In an geometry there are no similar figures. On the other hand, in relativity theory there is no kinematic similarity. If we multiply all components by a number k, then in the older theory the resultant will be k-times greater as well, however, not in relativity theory. We would have to draw either the plans of forces as well as all figures in absolute measure. However, as the unit length is too great, we can give in both cases only schematic, distorted images.

4. The Weierstrass coordinates.

As unit length we take 1 cm, that is the $$3\cdot10^{10}$$th part of the absolute unit distance of our an space. We want to measure the unit time so as to make the speed of light equal to one, i.e. in the resting medium light traverses the unit length (1 cm) in a unit time. We denote l as the new time that emerges from the old time by multiplication with the speed of light c as measured in the resting medium. We don't want to think of the considered clock as adjusted in the usual way, but as a simple clockwork that indicates, how often a certain (always repeating) process under the same circumstances, has happened since a certain event indicating the beginning of time counting. By that we always express the time indication of a certain clock by a single number l = ct.

An elementary event is defined by the value system x, y, z, l. I interpret the defining magnitude of that event or the variables x, y, z, l as homogeneous coordinates of a point in an three-dimensional space.

Let as first take the simple case that z = 0.

Through point M (Fig. 4) we lay two limiting arcs MT and MV normal to the coordinate axes. The lengths of these arcs and the hyperbolic cosine of the traveling ray OM are coordinates of point M. If we lay off from point M the