Today's equation in our march to Newton's birthday is actually a tiny bit out of order, historically speaking:
This is the Rydberg formula for the wavelengths of the spectral lines in hydrogen (and hydrogen-like ions), with R a constant having the appropriate units, and the two n's being two dimensionless integers. This equation was developed in 1888 by the Swedish physicist Johannes Rydberg (who was generalizing from a formula for the visible lines of hydrogen that was worked out by a Swiss schoolteacher, Johann Balmer). As such, it pre-dates Einstein's equation from yesterday, but its real importance in the history of physics comes later.
Rydberg's formula works because hydrogen has an exceptionally simple spectrum of emission and absorption lines, which allowed the pattern of lines to become obvious. Explaining this pattern was a great puzzle for physics at the end of the 19th century, and in fact, the answer wasn't worked out until 1913. In that year, Niels Bohr, who had been working with Rutherford around the time of his discovery of the atomic nucleus, first put forth the model of the atom that everybody nowadays learns in grade school.
Rutherford had the basic set-up before Bohr, of course-- a hundred years ago this year, he suggested a model of the atom with a positively charged nucleus orbited by negatively charged electrons. This model is flatly impossible in classical physics, though-- the moving electrons should produce radiation, which carries off energy causing them to spiral into the nucleus in a ridiculously short time. Given the physics known in 1911, there was no way to make the "solar system" model of the atom work.
Bohr's great leap in 1913 was, essentially, to throw everything that was known about physics in 1911, at least in certain special cases. He suggested that there are certain special orbits around the nucleus in which, for some reason, the electron will not produce any radiation. As long as the electron occupies one of these orbits, the atom is perfectly stable. Atoms absorb and emit radiation when they move between these special orbits, and the radiation they emit has a frequency determined by the same rule used by Planck and Einstein (though Bohr was weirdly resistant to the idea of light quanta, and spent a long time trying to cobble together an alternative in which light remained a classical wave).
Bohr suggested a simple condition for determining these stationary states-- that the angular momentum of the electron in one of these states was an integer multiple of Planck's constant divided by 2π (the famous h-bar of quantum physics). Using that condition, and known values of a bunch of fundamental constants, he could correctly predict the Rydberg formula for hydrogen.
While Einstein's particle model of light was a revolutionary step, it's nothing compared to Bohr's model of hydrogen, which really lacked any kind of justification other than the empirical fact that it worked. And it works brilliantly for hydrogen, or any atom stripped of all but one of its electrons. It fails for anything more complicated, but with the addition of some other ad hoc rules, you can cobble together a rough quantum theory that explains the spectra of other elements, though it's kind of a mess.
Bohr's model was the critical step that cracked the door open, though, and once the idea of quantized energy states of atoms was out there, other people figured out how to do it right. Louis de Broglie (from an aristocratic French family, his surname is nearly impossible to say correctly if you're not French) discovered that Bohr's model could be explained if the electrons behaved like waves, and this wave nature was soon demonstrated by the American physicists Clinton Davisson and Lester Germer and the British physicst G.P. Thomson (whose father J.J. Thomson won a Nobel prize for proving the electron was a particle). Soon after that, Werner Heisenberg, working with Bohr in Copenhagen, came up with a matrix theory that made correct predictions, followed soon after by Schrödinger's wave equation for the electron, and then Dirac's relativistic equation. It took 25 years to get from Rydberg to Bohr in 1913, but once Bohr got things going, the full quantum theory was worked out by 1930 or so.
So, take a moment today to appreciate the elegant simplicity of the most basic element there is, which provided the glimpse of an underlying pattern that got quantum theory rolling. And come back tomorrow to see the next equation in our series as we enter the final days of our countdown.