Split the light from a heated solid by wavelength with a prism and it comes out as a band of color running without a break. This is called a continuous spectrum.
Split the light from a discharge through hydrogen gas, however, and it does not come out as a band. Only a few thin lines stand at fixed positions. This is called a line spectrum.
Which positions the lines stand at is set by the kind of gas alone. For hydrogen they are at the same positions however many times they are measured. An atom emits light at particular frequencies only.
In the previous article a photon of h\nu = E_n - E_{n'} came out when an electron dropped a level. Rewritten in terms of wavelength this is \dfrac{hc}{\lambda} = E_n - E_{n'}.
If the levels are discrete, so are the differences. The wavelengths that come out are therefore discrete too, and the spectrum is lines rather than a band. A line spectrum is a direct sign that the levels are discrete.
The larger the difference, the shorter the wavelength. The difference is larger for a drop to an inner level, so those drops give light of shorter wavelength.
Group the lines that come out by the level they dropped to and they fall into several families. Such a family is called a series.
The lines dropping to n' = 1 are the Lyman series. The differences there are the largest, so all of them are ultraviolet. Those dropping to n' = 2 are the Balmer series, and these fall in the range the eye can see. Those dropping to n' = 3 are the Paschen series, which are infrared.
The first lines of hydrogen to be measured were the Balmer series. The family that happened to fall in the range the human eye can see was the one with n' = 2.
Putting E_n = -\dfrac{E_1}{n^2} into the formula for the difference gives \dfrac{hc}{\lambda} = E_1\left(\dfrac{1}{n'^2} - \dfrac{1}{n^2}\right).
Dividing both sides by hc gives \dfrac{1}{\lambda} = R\left(\dfrac{1}{n'^2} - \dfrac{1}{n^2}\right). Here R = \dfrac{E_1}{hc} is the Rydberg constant, 1.1 \times 10^7 per meter.
This formula had been found before Bohr, by looking at the measured wavelengths. The model of Bohr explained why that formula holds.
Take \dfrac{1}{n^2} along the horizontal axis and the lines of one series fall on a single straight line. The slope is -R, and the vertical intercept is \dfrac{R}{n'^2}.
Make n larger and \dfrac{1}{n^2} approaches 0. The lines crowd closer and closer together and approach one wavelength. This wavelength is called the series limit.
For the Balmer series it is 364 nm, and for the Lyman series 91 nm. Below that wavelength no lines stand, and a continuous spectrum takes their place. The electron has come away from the atom, and once away its energy is continuous.
The four the eye can see are red at 656 nm, blue-green at 486 nm, blue-violet at 434 nm, and violet at 410 nm. The way they crowd toward the short end comes from the way the levels crowd together.