The Periodic Table
In chemical elements with many electrons (like sodium), the outermost electrons could be made to jump from one energy level to another. Electrons located deeper down, closer to the nucleus, were however also expected to have definite energy levels. Arranging those elements in the order of the weight of their atoms (obtained from their density as gases, or in other ways) brought out two features. One, each had one electron more than the one preceding it (something on which x-rays also gave evidence), and two, some patterns seemed to repeat, again and again.
For instance, lithium, sodium, potassium--also rubidium and cesium--all had similar chemical behavior, consistent with a single active electron; the rest of the electrons formed in a tight spherical cluster centered on the nucleus. Such repeating patterns (quite a few exist) were first deduced in the 19th century by the Russian chemist Mendeleev ("Mendeleyev"), who used them to predict the existence of some yet-undiscovered elements.
The electrons remaining in the tight central "cores" of lithium, sodium, potassium etc. seemed to represent quite stable arrangements. The elements just ahead of them on the list--helium, neon and argon--were "noble gases" resisted any chemical combination, suggesting that all their electrons were in that core, in a bond so stable that no other atom could break into it.
What determined the number of electrons in that "core"--and therefore, the number of elements one needed to pass before a new "period" was formed? Lithium was the 3rd element--so its core had 2 electrons. Sodium 11th--meaning 10 electrons in the core, 8 more than lithium, probably as a "second layer" on top of the one of lithium (and so forth). By then the arrangement of energy levels was known from spectra, each with a number of "allowed" states of angular momentum, and the idea was that they formed "shells" in which allowed energy levels were completely filled. The scheme indeed worked, but only after Pauli suggested that each level accommodated exactly two electrons of opposite spins. Among heavier elements, sometimes shells filled in an order different from the one expected (some "outer" ones ahead of "inner" ones), but again, the pattern seemed sensible.
But only the pattern! One could perhaps classify the ordering of energy levels, but not predict their values. Nor could one predict the intensity of field lines--if an atom was at energy level "A", what were the relative probabilities of it jumping to "B" or to "C"? As one example--why was the transitions involving the double yellow line of sodium so prevalent that its light dominated all other emissions from that atom?
And if the atom seemed to resemble a miniature planetary system, why did the frequencies of the photons it emitted seem unrelated to the orbital frequency calculated for such planetary motions? And why the arbitrary rules about angular momenta? The Bohr-Sommerfeld theory seemed to give plenty of hints, but no precise answers.