In his book "Méthodes modernes en géométrie", Jean Fresnel wrote:
il ne faut pas se faire d'illusions, Descartes résout des problème de géométrie, non parce qu'il a de la méthode, mais parce qu'il a des idées
(Rough translation: "don't expect too much, Descartes solves geometric problems because he has ideas, not because he has a method".)
The context of this remark was that there are accessible reprints of La Géométrie that students could read for their benefit, but that they shouldn't have too high hopes. However, this casual remark of Jean Fresnel might still capture a general feeling in mathematics, namely that "content" (= ideas and technical details) is at least as important as "method" (= meta mathematics). However, there has indeed been unbelievable progress in mathematics in the time after Descartes. I wonder whether Descartes' "method" had something to do with it or not.
So here is my question: Has there been a Cartesian revolution in mathematics? And if yes, what exactly was so important about the work of Descartes?
Here is my own attempt at an answer: Descartes important contribution was his belief that there exists a "method" that simplifies geometry. This belief is in sharp contrast to the earlier belief passed down by Proclus Lycaeus (8 February 412 – 17 April 485 AD): "It is also reported that Ptolemy once asked Euclid if there was no shorter road to geometry than through the Elements, and Euclid replied that there was no royal road to geometry."