Annus MirabilisEinstein in 1905
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special relativity

The equation explained

β=11−(vV)2,\beta = \frac{1}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} ,
In wordsThe factor is one divided by the square root of one less the square of the speed of the moving system over the velocity of light.OverviewThe abbreviation, repeated under the final equations so that they can be read on their own.Full explanationThis is the same definition printed on p. 900, set again beneath the finished transformation. The identity worth carrying away from it is not the formula but what the square of the factor does to the bracket it is built from: β2(1−(vV)2)=1\beta^2 \left( 1 - \left( \frac{v}{V} \right)^2 \right) = 1. That is the step that makes every later calculation in the paper come out, in § 5's composition of velocities, in § 9's transformed field equations and in § 10's equations of motion. The factor is 1 at rest, about 1.25 at six tenths of the velocity of light, about 2.29 at nine tenths, and it has no value at all at v=Vv = V.Every stepβ2(1−(vV)2)=1\beta^2 \left( 1 - \left( \frac{v}{V} \right)^2 \right) = 1The square of the factor is the reciprocal of the bracket. This is the form in which the definition is used in the rest of the paper.v=0.6 V⟹β=11−0.36=10.8=1.25v = 0.6 \, V \quad \Longrightarrow \quad \beta = \frac{1}{\sqrt{1 - 0.36}} = \frac{1}{0.8} = 1.25The worked value this edition's laboratories use most.v=0.9 V⟹β=11−0.81=10.19≈2.29v = 0.9 \, V \quad \Longrightarrow \quad \beta = \frac{1}{\sqrt{1 - 0.81}} = \frac{1}{\sqrt{0.19}} \approx 2.29At nine tenths of the velocity of light. The factor rises slowly at first and then without limit.Historian's marginThe definition is printed twice in § 3, here and on p. 900, because the equations above it are printed twice: once with the unknown factor in them and once without. Keeping both displays is the edition's practice with the printed page, and the two records differ in what they are for. The earlier one introduces the abbreviation; this one states the identity the rest of the paper leans on.Also

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