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 the transformation is written with: one over the square root of one less the square of the speed ratio. It is 1 at rest and grows without limit as the speed approaches that of light.Full explanationThis is a definition, introduced to keep the four lines above short. The quantity it abbreviates is the one that appeared by itself in the algebra: VV2−v2\frac{V}{\sqrt{V^2 - v^2}}, which is the same number written without dividing through by VV. The paper writes it β\beta, and it is the letter the whole paper then uses, in §§ 4, 6, 7, 9 and 10. It is never smaller than 1, it is 1 exactly when the system is at rest, and it has no value at all once vv reaches VV, which is one of the places where the theory puts a limit on the speeds it admits.Every stepVV2−v2=VV1−v2V2=11−(vV)2\frac{V}{\sqrt{V^2 - v^2}} = \frac{V}{V \sqrt{1 - \frac{v^2}{V^2}}} = \frac{1}{\sqrt{1 - \left( \frac{v}{V} \right)^2}}Take VV out of the square root and cancel it against the one in front. The two forms are the same number.v=0⟹β=1v = 0 \quad \Longrightarrow \quad \beta = 1At rest the transformation reduces to the identity in this factor, as it must.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} = 1.25A worked value, the one the laboratories of this edition use most.Historian's marginThis factor is not Einstein's invention and not his notation. It is the one Lorentz's transformation of 1904 already carries; what § 3 has done is to derive it from the two principles stated in § 2 rather than assume it. Later writing calls it γ\gamma, and the concordance of this edition marks the collision, because the paper's own β\beta is the modern γ\gamma, not the modern β\beta, which is the speed ratio v/Vv/V.Also

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