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In the hydrogen spectral series how did the scientists know the number of the energy level which the electron is moving from or to?

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The numbers are not measured, they are part of a model that explains why the hydrogen atom emits/absorbs only at those wavelengths.

The first series of Hydrogen lines to be discovered was the Balmer Series, and nobody knew why they were discrete lines instead of a continuous spectrum. Johann Balmer discovered that the lines all had wavelengths equal to, $$\lambda_n = \left({n^2 \over n^2-4}\right)\times 364.5068\;\text{nm.}$$

Later on, it was realized that there were other lines with $m^2$ instead of 4 in the denominator. Later still, the Bohr model of the atom was able to reproduce this pattern of energy levels. Improvements to this came with Quantum Mechanics and various corrections.

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