They were mimicking beverages made by fermentation.
Mild carbonation is a byproduct of the fermentation process. In addition to alcohol, yeast gives off carbon dioxide which dissolves in the beverage. When the cask or bottle is opened, the carbon dioxide comes out and creates the classic fizz.
Folks who were looking to recreate the effect in other beverages (like water) used the carbon dioxide released during the fermentation of beer to impregnate their beverage with fizz. As a result, soda water ended up using carbon dioxide too. Later techniques used bicarbonate of soda and tartaric acid to get the fizz, and by chance, the fizz produced by mixing those two chemicals is _also_ carbon dioxide.
Modern processes use pressurized carbon dioxide rather than a chemical process, but at this point, CO2 is just the standard used for making stuff fizzy. Some beverages use other gasses (Guinness, most famously, uses nitrogen) but that requires more expensive production and storage techniques.
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