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"From reasons of symmetry it is now evident that the length of a given rod moving perpendicularly to its axis, measured in the stationary system, must depend only on the velocity and not on the direction and the sense of the motion. The length of the moving rod measured in the stationary system does not change, therefore, if

*v*and

*−v*are interchanged. Hence follows that

*l/φ(v) = l/φ(−v)*, or

*φ(v) = φ(−v)*."

As it seems to me these reasons of symmetry are basically assumption that there is no preferred reference frame and transformations between different reference frames should be completely symmetric without any reference to some absolute state of motion.