TITLE: Was there some prior idea that inspired both Fermat & Descartes to invent coordinates?
QUESTION [15 upvotes]: It seems incredible to me that both Descartes & Fermat could have both simultaneously discovered such a novel & significant idea, without there being some single prior idea that they both could have taken inspiration from. Has there been any research done on this, or can someone expliciate the histrorical record further?
Wikipedia does state that Nicole Oresme in the 14C made constructions similar to coordinates. This is well before either Descartes or Fermat. It doesn't state whether they were influenced by him.
Alternatively, could they have been influenced by developments in Cartography or Map-making?

REPLY [3 votes]: To complement Henning Makholm's fine answer, I would add that it is not entirely accurate to say that "The improved notation had been in development for at least a couple of centuries, with algebraists gradually freeing themselves from the classical/medieval tradition of describing everything in prose." Here the key name is Vieta.  His work introduced the idea of symbolic mathematics in a systematic way, and constituted the transition to the modern period as far as symbolism is concerned. Both Fermat and Descartes relied on this.