$$2\pi r, \, \pi r^2,\, \frac{4}{3}\pi r^3\ldots$$
We all know these formulas, and we all have some idea about how they were introduced by some hard-thinking Ancient Greeks.
We also picture the proofs of such formulas as being purely geometric, and somewhat complicated and un-enlightening.

I will explain how the mysterious aspect of these proofs amazed Philosophers and Mathematicians in the first place,
and how the lack of method inspired more modern Mathematicians such as Descartes, Fermat and Co. to build their own method and shape the Infinitesimal Calculus we use today.

This is intended as a short informal survey, but I hope to show you some less-known aspects of the History of Mathematics.
No particular knowledge is required.