How is Cartesian coordinate system related to his philosophy

Question

In 1637, Rene Descartes published his famous monograph about philosophy "Discourse on the Method of reasoning well and Seeking Truth in the Sciences", and analytic method of geometry has been come up with in the appendices. I want to know is there any philosophical thinking or method leading to his invention of coordinate system? Which philosophical rule led him to connect numbers with geometry?

Answer 1

For a detailed but very readable discussion of this and related questions about the connection between Descartes' philosophy and his mathematics, and about the wider historical context, see the freely available

Stanford Encyclopedia of Philosophy entry on Descartes' Mathematics

The general standard of SEP entries, and the standard of entries related to logic and the philosophy of mathematics in particular, is extraordinarily high: and though I'm no historian, this strikes me as no exception.

Answer 2

For more details, you can supplement SEP's entry with the book-lenght study :

Chikara Sasaki, Descartes' Mathematical Thought (2003).

All first part (about 150 pages) is devoted to the historical background of D's studies (which forms the bulk of SEP's entry) but you can see Ch.4 : THE MATHEMATICAL BACKGROUND OF THE REGULAE AD DIRECTIONEM INGENII (page 159-on).

This unpublished work contains a (failed) attempt to develop a Mathesis universalis, i.e. a general method to "analyze" all problems, and not only mathematical ones [see Sasaki, page 190-on].

Ch.5 of Sasaki's book is devoted to : THE GéOMéTRIE OF 1637 [page 205-on].

Clues of the hidden links between the "thin" rules of method of the Discours de la méthode and the Géométrie must be searched into the Regulae.