There are seen four lines: red, blue and two violet (and another violet seen only by students with enough extended immagination1) ).
It was needed a ordinary Lyceum teacher (and from 1865 the assistant in Geometry at Bassel University) to find out a dependence of wavelength λ from squares of numbers 1/22, 1/32 etc.
1/λ=R’(1/22
– 1/n2),
with n≥3 and R related to so called Rydberg's constant (or ionisation energy of hydrogen: R=13,6 eV )
A bit of luck (or rather a help from the human evolution process, which made our eyes more sensitive to this wavelength range) - the Balmer's series is due to transitions between second (i.e. the first excited) orbit and higher orbits (i.e. vice versa). It is not a "fundamental" series, i.e. transitions to the first orbit, but that one (Lyman's 1905) falls in ultraviolet.
Another transitions, to higher orbits in range of infrared were discovered first in spectra of the stars and then in laboratory (n = 4, Pashen, 1908 r.; n = 5, Brackett - 1922 r.; n = 6, Pfund - 1924 r.).
1)In fact, Physics students know well that Balmer's lines are only a fantasy, because without them the lab work will not be approved. In real spectra of the discharge pipe that entire colorful belt is seen in spectrometer window and without help from older collegues (which show younger students what the professor wants) no line would be really visible.
„Visible” means that range of wavelengths is in octave of "visible" light: 380-760 nm.
