Tuning the Atoms 251
squares of the numbers 3, 4, 5, 6, and as denominator one of these less one of the four
smaller numbers [i.e., the squares of 1, 2, 3, 4]. ”^16 Once having set out to find the funda-
mental number, and using the Pythagorean presumption that it would be multiplied by
simple fractions to give the observed hydrogen lines, Balmer was then led to his coeffi-
cients, thence to their numerators (which he realized could all be expressed as squares of
integers), making the inference that their denominators were the differences of squares a
fairly small further step.
Nor would Balmer have simply expected that his coefficients all be simple Pythagorean
ratios, though two of them (4/3 and 9/8) indeed were; his reading of Helmholtz would
have assured him that more complex bodies than simple vibrating strings have more
complex overtone structures, yet are still governed by the same mathematical principles.
For instance, Helmholtz reviewed the harmonic series of a vibrating circular plate, already
famous from Chladni ’ s experiments and familiar to mathematicians via the special func-
tions devised by Friedrich Bessel describing their modes of vibration.^17 In this case,
Helmholtz noted that “ there is no commensurable ratio between the prime tone and the
other tones. ” Balmer could also have found precedent for his use of squared integers in
Chladni ’ s empirical law, which expressed the frequency of vibrating bodies as roughly
proportionate to the squares of a series of integers.^18 Balmer was probably pleased and
surprised that the vibrations of hydrogen seem comparable in complexity with those of a
circular plate.
After obtaining these results, Balmer learned from his friend Eduard Hagenbach about
the new spectroscopic discoveries of Edward Huggins and Hermann Vogel, which he
mentioned at the beginning of his paper. Evidently, in hot white stars hydrogen reach states
capable of producing lines not heretofore observed in earthly experiments. Where Stoney
had merely noted the four Å ngstr ö m lines as “ overtones, ” without considering other pos-
sibilities, Balmer ’ s formula led him to consider the potential infinitude of spectral lines as
the integers m and n take on ever larger values. Already in his first publication, Balmer
explored this possibility and found good agreement with the newly observed lines ( figure
16.4 ). Although he did not go further still to use his formula to predict as-yet unobserved
lines, Balmer ends by proposing that his general approach could be extended to other ele-
ments, which he discussed in more detail in his fourth and final publication (1897).^19
Though Balmer ’ s initial publication was in a Swiss journal (published in his hometown,
Basel), when he republished these same results later that year in a well-known German
journal, Annalen der Physik , he omitted all the explanatory material that shows the con-
nection of his work with the theory of overtones.^20 Instead, his formula appears to come
from nowhere, as if plucked from the ether by pure mathematical speculation; this expur-
gated version of his discovery conforms to what was increasingly a preference to erase
the context of discovery from the published account. In the case of acoustics, music tended
to recede further into the background as time went on. Whereas in Helmholtz ’ s Tonemp-
findungen music figured very prominently, a decade and a half later Lord Rayleigh ’ s