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Johann Balmer (1825–1898), a Swiss mathematician, found empirically in 1895 that the separation of the optical lines generated by hydrogen gas can be expressed by a formula using just a constant, C, and integer numbers. He expressed his observation with Eq. (1.3)

(1.3)

here λ is the wavelength of the missing line, C is a heuristically obtained constant (C = 3.64 × 10⁻⁹ m), n = 2, and m is an integer greater than 2 (i.e. 3, 4, 5, and so on). When you put any of these integer numbers in Eq. (1.3), you get the wavelength of all the lines in the hydrogen spectrum.

Figure 1.6 The spectrum of the hydrogen atom on the left shows the absorption lines (below) and the emission lines (middle). On the right are the emission lines of several other materials.

Source: https://www.shutterstock.com/image‐vector/spectrum‐spectral‐line‐example‐hydrogen‐emission‐1288942888?src=iUiOwiDEznOcV6XzswXhMA‐1‐0 (left); https://www.shutterstock.com/image‐vector/line‐spectra‐elements‐339037577?src=I6tWF1qlh6XcWayXsZl‐Gw‐3‐16 (right).

Figure 1.6 shows the hydrogen spectrum on the left, with its characteristic emission and absorption lines. These are the lines that Balmer used to develop Eq. (1.3) to calculate the missing hydrogen's wavelengths. All the elements have similar absorption and emission lines at different wavelengths, and I show a few on the right in Figure 1.6.

Just three years later, Johannes Rydberg (1854–1919) found that the Balmer equation was one specific case of a more general formula, Eq. (1.4):

(1.4)

The reciprocal of the wavelength is now given by a constant R and the same integer numbers, except that now n is allowed to have different integer numbers: 2, 3, 4, and so on. R is also a heuristically derived constant (R = 1.1 × 10⁷ m⁻¹), called the Rydberg constant. Both Balmer and Rydberg (Figure 1.7) were able to quantify the entire spectrum of the hydrogen atom using the relationship in Eq. (1.4). It is interesting that Niels Bohr, whom I'll talk more about in Section 1.8, was able to calculate the Rydberg number using fundamental physical values, such as the mass of the electron, the electronic charge, the permittivity of free space, Planck's constant, and the speed of light (see Appendix 1.3). This behavior screams for an explanation.

Figure 1.7 Johann Balmer (left) found a mathematical relation for hydrogen's spectral lines, and Johannes Rydberg (right) came up with a more general equation applicable to all gases and materials.

Source Wikipedia, https://en.wikipedia.org/wiki/Johann_Jakob_Balmer#/media/File:Balmer.jpeg (left); Wikipedia, https://en.wikipedia.org/wiki/Johannes_Rydberg#/media/File:Rydberg,_Janne_(foto_Per_Bagge;_AFs_Arkiv).jpg (right).