What I am referring to is the Balmer formula as it appears in Wikipedia. To come up with this series by trial and error along with its constants is asking a little too much but I can't understand how one would go about deriving this formula which involve such exquisite measurements of its constants. Specifically was there some perhaps experimental methodology that makes the formula a little more easy to see how he may have arrived at it? It appears to be an ad hoc truly amazing leap in intuition. Which may be the case. Certainly doesn't hurt to ask.
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Here the German Wikipedia helps better than the English one: Balmer was interested in a very broad range of sciences and "sciences" including numerology and cabbala. For instance he calculated the number of steps of pyramids and the floor plan of Jewish temples, probably from the information supplied by the bible. Therefore it is not surprising that he was interested in the problem of spectral wavelengths which Eduard Hagenbach (Prof. of physics at Basel university), knowing Balmer's deciphering skill, had suggested to him. In 1885 Balmer found the well-known formula
$\lambda = \frac{m^2h}{m^2-n^2}$ where $n = 2$ and $m = 3, 4, 5, ...$
which in 1888 was generalized by Rydberg.
Balmer predicted the line for m = 7 which was confirmed by Angström. Other lines, i.e. higher quantum states, were found in the spectra of white stars.
Perhaps also of interest, but only available in German from the Swiss Mathematical Society: Balmer also lectured as a Privatdozent at the university of Basel. Alas there were no students interested. Of about 70 announced courses 50 did not take place. Only very rarely he had more than 1 or 2 students. So his main job remained to teach girls how to add and multiply fractions and to express thoughts in written form, 17 hours a week.
In his notes of 1891 he speculates about atomic movement, points of force in circular or elliptic motion and even something resembling waves. These (historically) interesting notes are partially available on p. 59.
