 Albert Girard

Albert Girard studied relations between the roots and coefficients of polynomials. His 1629 Invention nouvelle en l'algèbre treated sums of roots, products of roots and sums of their powers, connecting equation solving with what became the language of symmetric polynomials.

For a monic quadratic with roots $r,s$, the identity $(x-r)(x-s)=x^2-(r+s)x+rs$ makes the connection transparent. Girard's work extended this viewpoint beyond a single quadratic and included roots that could not be treated as ordinary positive magnitudes.^[ref]