The Cartesian Method: Mathematical or Geometrical?

In his work, Rene Descartes has referred to the effect of mathematics on his mind. He, as a founder of algebra and analytic geometry, believes that it is only the mathematical knowledge that can properly be called “knowledge”. However, the question is why the Cartesian method is referred to by some of his commentators as well as the majority of Persian sources as a “geometrical” method, while most scholars have emphasized on his mathematical method. Descartes paved the ground for mathematical physics. For him, geometrical features such as point, dimension, and extension are essential to the matter. I believe that the emphasis on his geometrical method and the priority of geometry amounts to the priority of the material substance and materialism in Descartes, which is in conflict with Cartesian views, because he believed in souls and God as two immaterial substances. In his philosophy, mathematics is originated in the reason and rationality, and mathematical proofs are certain and clear just because of their rational origins. However, the mathematics involved in the mathematical method is not the general ordinary mathematics; rather it is the mathematics that was divided, since the time of Aristotle, into “pure mathematics” and “applied mathematics”. This paper seeks an answer to the following questions: what is the difference between the geometrical and mathematical methods? In which of the two is there more clarity, distinctness and certainty? For Descartes, what problems can be solved by the mathematical method and what problems can be solved by the geometrical method?