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In 1666 Isaac Newton argued that the spread of light by a prism showed that light was in fact constituted of many colors from violet to red, passing through blue, green and yellow.

In 1800 William Herschel finds more heat in the dark region past the red than in the red itself, thereby discovering the existence of "invisible light" in the infra-red region. A year later Johann Wilhelm Ritter discovers the existence of potent "invisible light" in the the dark ultra-violet region. Another year later, William Wollaston mentions the existence of dark bands in the color spectrum of sunlight.

It was another 12 years, in 1814, before the young Joseph von Fraunhofer rediscovered these lines and began a detailed study of spectral lines from various sources. By the time he died in 1826, at the age of 39, the existence of these lines was well established, but the nature of their origin was still hidden.

From 1826, progress on the study of spectral lines and how they relate to chemical elements was rapid. John Hershel (son of William), Fox Talbot, David Brewster, William Miller, Edmond Becquerel, Jean Foucault all made important contributions.

Then, in 1859, Gustav Kirchhoff and Robert Bunsen reached a milestone: they explained the relation between absorption lines (dark) and emission lines (bright). They showed that every element, no matter what compound it is in, gives the same spectrum. Spectral analysis was shown to be a simple and powerful tool for identifying elements (and by now, atoms). In 1860 Kirchhoff and Bunsen discover cesium, then a year later, rubidium, purely from spectral analysis [1].

It is worth remembering that up to this point (1860), and for next 50 years, why every element has a signature spectrum is unknown. It is just established by observation that it is so.

In 1869 the astronomer Joseph Norman Lockyer observes an unknown bright yellow line in the solar spectrum, and in spectroscopy's most spectacular achievement, discovers helium (greek helios, sun) in the Sun 25 years before it is identified on Earth (Ramsay, 1895) [2, 3].

Line spectra could not be explained by the physics of the 19th century. In 1913, Niels Bohr, building on the quantum idea pioneered by Max Planck (black body spectrum, 1900 [4]), hypothesized that atoms can only exist in discrete stationary states. This was enough for him to derive in detail all the lines in the hydrogen spectrum. By the time quantum mechanics had become an acceptable theory, in the mid 1920's, it preserved Bohr's insight whole: Atoms have discrete stationary states that can be calculated from our understanding of electromagnetism. The light emitted or absorbed by an atom corresponds to transitions between these states.

[1] Spectrum History, John Park's ChemTeam: Classic Papers from the History of Chemistry (Cached)
[2] Helium's 100th Aniversary, chemmybear.com
[3] Trail of the Elements(broken), James R. Fromm, Imbris, Inc (Cached).
Plank's Formula for Black Body Radiation, American Physical Society.