  It may seem a funny notion to write about theorems as old and rehashed as
Descartes's rule of signs, De Gua's rule or Budan's. Admittedly, these theorems
were proved numerous times over the centuries. However, despite the popularity
of these results, it seems that no thorough and up-to-date historical account
of their proofs has ever been given, nor has an effort been made to reformulate
the oldest demonstrations in modern terms. The motivation of this paper is to
put these strongly related theorems back in their historical perspective. More
importantly, we suggest a way to understand Descartes's original statement,
which yet remains somewhat of an enigma. We found that this question is related
to a certain way of counting the alternations and permanences of signs of the
polynomial coefficients, and may have been the convention used by Descartes.
Remarkably, this convention not only provides a ultra-simple proof of
Descartes's rule, but it can also be used to simplify the proofs of the titular
theorems. Without claiming to be exhaustive, we shall present in this paper an
historical account of these theorems and their proofs, and clarify their mutual
relation. We will explain how a suitable convention can help understand the
original statement of Descartes and greatly simplify its proof, as well as the
proofs of the above-mentioned theorems. With the exception of the proof of
Fourier's theorem and its generalizations, which run on rudiments of
infinitesimal calculus (Taylor's theorem), the proposed demonstrations are so
short and elementary they could be taught at the undergraduate level.
