Annus MirabilisEinstein in 1905
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special relativity

The equation explained

ξ=Vτ,\xi = V \tau ,
In wordsThe distance along the motion in the moving system is the velocity of light times the moving system's time.OverviewA ray sent along the moving system's first axis must be found, by that system's own measurements, to travel at the velocity of light. That is this line, and it is where the second principle enters the derivation of the distances.Full explanationThe paper has the moving system's time and now wants its distances. The requirement it uses is the one the two principles together demand: light, measured in the moving system, also travels at VV. So a ray sent from the origin at τ=0\tau = 0 along the direction of increasing ξ\xi is at ξ=Vτ\xi = V \tau at the moving system's time τ\tau. The line is short but it is not an identity: it is the constancy of the velocity of light, asserted now of a system the paper has not yet described, and it is what the rest of the section has to be consistent with. Where the first displays used the velocity of light in the stationary system to fix the two legs of a signal, this uses it in the moving one.Every stepdistance=speed×time\text{distance} = \text{speed} \times \text{time}For a ray that leaves the origin at τ=0\tau = 0 and travels along one axis at a constant speed.ξ=Vτ\xi = V \tauThe speed measured in the moving system is VV, by the principle of the constancy of the velocity of light taken with the principle of relativity, which is what the paragraph states.Historian's marginThis is the step the paper's title paragraph promised and the one that separates it from an account in which the moving system's clocks are merely mis-set: the same VV is required in both systems, rather than derived in one and carried to the other (p. 899).Also

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