Marin Mersenne published L’Harmonie Universelle in 1637.
The work of Pythagoras and Vincenzo Galilei, of course, was earlier.
Pythagoras died around 495 BCE, and Galilei died in 1591.
Some sources say that Galileo Galilei,
independently of Mersenne and about the same time,
discovered all three laws of vibrating strings.
By “a non-linear mathematical equation” we mean that the equation
relates two things that are not merely directly or inversely proportional to each other,
but that depend on the square or square root, cube or cube root.
See also in The book of science:
Readings on wikipedia: