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Short answer: Descartes' doubt was like WWI - the doubt to end all doubt. Only by doubting everything could Descartes hope to find anything that was certain (even if the only certainty is that nothing else is certain!).
Analytic geometry.
Who inspired René Descartes? In Breda, Descartes was encouraged in his studies of science and mathematics by the physicist Isaac Beeckman (1588–1637), for whom he wrote the Compendium of Music (written 1618, published 1650), his first surviving work.
Rene Descartes 1596-1650, a Frenchman who spent most of his living days in the former Dutch Republic. Descartes who is also known as the Father of Modern Philosophy due to much of his work still heavily influencing philosophy today.
Descartes did not invent polynomials.