Annus MirabilisEinstein in 1905
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The equation explained

1T=−1βνlg⁡ϱαν3\frac{1}{T} = -\frac{1}{\beta \nu} \lg \frac{\varrho}{\alpha \nu^3}
In wordsOne over the absolute temperature equals minus one over the constant β times the frequency, times the natural logarithm of the energy density divided by the constant α times the frequency cubed.OverviewWien's law solved for one over the temperature. Section 3 showed that 1/T is the rate at which the entropy density grows with the energy density, so this is the rate the next step integrates.Full explanationTake the logarithm of Wien's law and solve it for 1T\frac{1}{T}; the paper writes lg⁡\lg for the natural logarithm. The result expresses the temperature through the density at one frequency. Where the density is small compared with αν3\alpha \nu^3 the logarithm is negative and 1T\frac{1}{T} positive, as it must be. Section 3 found that in black radiation ∂φ∂ϱ=1T\frac{\partial \varphi}{\partial \varrho} = \frac{1}{T}, so this line is the rate at which the entropy density grows with the density, ready to be integrated.Every stepϱαν3=e−βνT\frac{\varrho}{\alpha \nu^3} = e^{-\beta \frac{\nu}{T}}Divide Wien's law by αν3\alpha \nu^3.lg⁡ϱαν3=−βνT\lg \frac{\varrho}{\alpha \nu^3} = -\beta \frac{\nu}{T}Take the natural logarithm of both sides, which the paper writes lg⁡\lg. The logarithm of an exponential is its exponent.1T=−1βνlg⁡ϱαν3\frac{1}{T} = -\frac{1}{\beta \nu} \lg \frac{\varrho}{\alpha \nu^3}Divide both sides by −βν-\beta \nu: the printed line.Also

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