Volker Thomsen

Although spectral lines had been observed since the early years of the 19th century, it wasn't until 1860 that Kirchhoff and Bunsen established them as unique to each element. In 1868, Swedish physicist Anders Jonas Ångström (1814–1874) published a detailed study of the wavelengths of solar spectral lines, expressed in units of 10^{–10} m. This unit is now known as the angstrom (Å).
Johann Balmer
The Swiss mathematician Johann Jakob Balmer (1825–1898) studied at the universities of Karlsruhe and Berlin and earned his doctorate in 1849 from the University of Basel. Beginning in 1859, he taught mathematics at a secondary school in Basel. In 1865 he also became a university lecturer in mathematics at the University of Basel. His main field of interest was geometry. He was persuaded by university colleague E. Hagenbach to search for a mathematical formula to properly characterize the hydrogen spectrum. Previous attempts had failed.
But Balmer succeeded. In 1885 he showed that the wavelengths (λ) of the visible spectral lines of the element hydrogen could be represented by a simple mathematical formula (1):
Table I: Balmer's formula versus measured values

where n = 2 and m = 3, 4, 5, . . . He called his constant h = 3645.6 Å "the fundamental number of hydrogen." Balmer's empirical formula provided very accurate wavelength values for n = 2 and m = 3, 4, 5, and 6 as shown in Table I.
In his paper, he predicted a wavelength for m = 7, and in the same paper notes that he was informed by his colleague at the University of Basel (Hagenbach) that this spectral line had been observed and that the wavelength was in excellent agreement with his prediction. He also showed that his formula accurately produced the wavelengths of several other hydrogen spectral lines.
Balmer also suggested that other spectral line series for hydrogen might be found using other small integer values for n. For those interested, his original (translated) paper is a must read. It is also interesting to note that Balmer was 60 years old when he made this discovery.
Johannes Rydberg
In 1888 the Swedish mathematician and physicist Johannes Rydberg (1854–1919) reformulated Balmer's equation as
where a is a constant, n
_{1} and n
_{2} are integers, and n
_{2} > n
_{1}. The constant is now called Rydberg's constant for hydrogen (R_{H}) and has a value of 1.097 × 10^{7} m^{–1}. For the Balmer series of hydrogen lines n
_{1} = 2 and n
_{2} ≥ 3 in this equation.
It is unclear from the historical literature whether Rydberg knew of Balmer's previous work. A recent biography (2) notes that he was independently pursuing research on this problem when he learned of Balmer's work. The authors suggest that this merely helped confirm the direction that he was already pursuing. The Rydberg formula is, however, easily derived from that of Balmer (Appendix 1). Nevertheless, Rydberg's formula pointed the way to other spectral lines series for hydrogen as Balmer had speculated. Rydberg also paved the way for further mathematical clarifications of higher atomic number elements.