When was a mathematical formula (instead of just words) used for the 1st time in natural sciences to describe a natural phenomenon?
Pythagoras discovered that simple mathematical ratios were important to harmony. As Xenocrates put it:
Pythagoras discovered also that the intervals in music do not come into being apart from number; for they are an interrelation of quantity with quantity. So he set out to investigate under what conditions concordant intervals come about, and discordant ones, and everything well-attuned and ill-tuned.
This is reported over a century after the discovery. Any earlier such discovery has not come down to us.