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coordinate transformation as indicated in Fig. 4, also the uniformity of motion is canceled.

According to Minkowski the world line illustrating uniform motion, is the straight line in the xt-plane trough the coordinate origin.

In this connection the following thought of Palagyi^[1] is interesting: "What we consider as the path traveled by the material point within a space imagined as stationary, is only the apparent path of the point; the real path results from the construction of the right angled parallelogram of the apparent path and the elapsed time ... This had to be a consequence of the fact, that even the state of rest was interpreted by us as uniform motion, and specifically as a motion in the subjective dimension of time. Thus by the parallelogram we put together this subjective component with the objective component that is given by the apparent path, and thus we get the real path traveled by the point in streaming space.
This real way together with the time ray constitutes an angle φ, that I call the time angle of uniform motion."

The velocity was expressed by him by the tangent of time angle φ.

The image of accelerated motion is the distance line. To its intersection with the ordinate axis we attribute the velocity 0; in its additional points we have

$v=c\ \operatorname {th} \,\ X$

The velocity increases together with the distance up to the velocity of light.

5. The principle of Doppler

In my first work on the non-euclidean interpretation of relativity theory^[2] I transformed the first formula of Einstein for the Doppler principle into the form

${\frac {\nu '}{\nu }}={\frac {\operatorname {ch} \,\left(u_{1}-u\right)}{\operatorname {ch} \,u_{1}}}$

(28)

We take two equidistant lines (Fig. 6) with the X-axis as their common center line and with the parameters $OA=u_{1},\ OA'=u_{1}-u.$. The arcs of those two distance lines limited by the Y-axis and the perpendicular to the X-axis in point C, are

$AB=OC\ \operatorname {ch} \,u_{1},\ A_{1}B_{1}=OC\ \operatorname {ch} \,\left(u_{1}-u\right)$

Thus it is

${\frac {\nu '}{\nu }}={\frac {A_{1}B_{1}}{AB}}$

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The relation between frequencies $\nu '$ and $\nu$ can be illustrated in the general case as the relation of the arcs of two distance lines between common perpendiculars.

For φ = 0 we have $u_{1}=\infty$. The distance lines go over into the limiting circles with the common axes, and we have

$\nu '=\nu e^{-u}$

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Fig. 7 corresponds to that case.

In euclidean geometry the distance lines and the limiting circles are reduced to parallels to the given straight line. The expressions for Doppler's principle in ordinary mechanics can be easily illustrated by intersections of parallel transversals between the legs of an angle.

6. Aberration

The light ray T strikes the X-axis under the angle φ. To find the direction of the deflected light ray, we have to draw a Lobachevskian parallel from N to T. The aberration equation is

$u'_{1}=u_{1}-u$

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The construction of the needed parallel is executed in Fig. 8 as follows.^[3]

1. Palagyi, Neue Theorie des Raumes und der Zeit, 1901, p. 45.
2. This journal. 11. 93, 1910
3. Lobatschevskij-Engel, Zwei geometrische Abhandlungen, p. 256