Final answer:
Balmer's formula for hydrogen spectral lines and Rydberg's generalization were both derived empirically by fitting them to experimental data, the latter including a constant and predicting lines beyond the visible spectrum. The exact positions of spectral lines could be given by these formulas before a theoretical justification was established in quantum mechanics.
Step-by-step explanation:
Balmer constructed his formula for the hydrogen spectral lines by trial and error manipulations of large amounts of experimental measurement data. This is somewhat like the procedure followed by Johannes Rydberg, who generalized Balmer's work and developed an empirical formula that predicted all of hydrogen's emission lines, not just those visible to the eye. The Rydberg formula included a constant now known as the Rydberg constant and took into account different series of lines observed in the ultraviolet and infrared regions, beyond Balmer's original formula which only described the visible lines emitted by hydrogen atoms.
Johann Balmer, who was a mathematics teacher, derived his equation empirically by fitting it to experimental data. This approach led to the discovery of a pattern in the wavelengths of light emitted, which paved the way for the future development of quantum mechanics. Rydberg extended this work and his formula is significant for giving the exact positions of the spectral lines as observed in laboratories. However, both Balmer's and Rydberg's formulas were initially without theoretical justification, a problem that was not resolved until the early 20th century with advancements in atomic theory.