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When was a mathematical formula (instead of just words) used for the 1st time in natural sciences to describe a natural phenomenon?

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    $\begingroup$ There is no "first", as is often the case, what counts as "mathematical formula" is too vague for that. Modern symbolic notation was only established in the 17th century, before that various mixtures of symbols and words were used. Already Euclid and Archimedes used letters and proportions to express geometric relations in diagrams that described optical or mechanical phenomena, and they already followed an existing tradition. So one can pick anything between Pythagoreans and Descartes for the "first" depending on what "formula" means. $\endgroup$
    – Conifold
    Feb 2 at 0:58
  • $\begingroup$ Modern "algebra-like" formulas dates from Late Renaissance-Early Modern: Descartes. See here for a Sixteenth-century example, how it is difficult to read the eqaution without formulas. $\endgroup$ Feb 2 at 8:16

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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.

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