Measuring the arcs of the orbit of a one-parameter transformation group

In the orbits of the one-parameter transformation group, various geometric structures are introduced by means of bijection with the set of real numbers, in particular, the structure of the Euclidean straight line in the sense used by Hilbert with corresponding concepts: points, half-orbits (an analogue of a ray), an oriented arc (an analogue of a segment with ordered endpoints), the relations of “between” for three points, the equality of oriented arcs and other invariants of the transformation group. It is shown that the measure of arcs, defined as a positive definite additive function that is invariant with respect to the group of transformations, exists and splits into two independent measures which are uniquely defined on the classes of arcs of a similar orientation by setting the standards — one in each class — and coinciding with the measurement results by these standards from different endpoints of the arc. $\lambda$-Congruence permits to measure oppositely oriented arcs using a single standard. In this case, the opposite arcs $ab$ and $ba$ (not $\lambda$-congruent) have different measure values. This circumstance forces us to question the correctness of the known proofs of the existence and uniqueness of the measure of the length of a line-segment in Euclidean space. With $\lambda$-congruence for $\lambda = -1$, the orbit becomes a model of Euclidean straight line.