Note 66 — Chapter XIX · VARIOUS POSSIBLE WORLDS
Reading edition
The general expression giving the composition of velocities in all three cases can be understood as follows: If a ball is moving through a train with velocity as measured in the train, and if the train is moving with a speed in the same direction along the embankment, then the speed of the ball with respect to the embankment is given by , where has the following expression: In this formula represents the value of the invariant velocity in that particular type of world we are considering.
In classical science, is infinite, and the formula degenerates into . This is the well-known classical composition where velocities lying in the same direction are added algebraically.
In Einstein’s theory, is equal to Maxwell’s constant . This being a real finite number, we obtain the Lobatchewskian type of composition. Were an imaginary number, we should obtain the Riemannian type.
We may note that if for or we substitute , that is to say, if we wish to discover what would result from the addition of a velocity to the invariant velocity, we obtain This proves that the addition of a velocity to the invariant velocity still leaves us with the invariant velocity, justifying thereby the maximum and invariant nature credited to .