What is analytic geometry? How does Descartes's method contribute to his discovery of analytic geometry?

What is the significance of Descartes claim, "I am thinking, therefore I exist"? How does he argue for that claim?

What are Descartes's two proofs of God's existence? What are the differences between them? Is one better than the other? Do you find them satisfying?

Why does Descartes present his physical theories as an imaginary hypothesis? Does this hurt their plausibility?

Explain the difference between a Cartesian physics based on a definition of matter and a Newtonian physics based on a definition of force. Which is more successful and why?

What reasons does Descartes give for not making the principles of his physics public? What do you think are the real reasons?

To what extent do Descartes's writings reflect the medieval tradition in which he was educated, and to what extent do they reflect a new, scientific worldview?