  In his \textit{Brouillon Project} on conic sections, Girard Desargues studies
the notion of \textit{traversale}, which generalizes that of diameter
introduced by Apollonius. One often reads that it is equivalent to the notion
of \textit{polar,} a concept that emerged in the beginning of 19th century. In
this article we shall study in great detail the developments around that notion
in the middle part of the \textit{Brouillon project.} We shall in particular
show, using the notes added by Desargues after the first draft was written, how
here arises a new \textit{theory,} that of the \textit{polarity} associated to
a conic section. We shall show that besides a natural correspondance between
points and lines, Desargues has understood that a conic induces on every line
in its plane an involution, and how he progressively handles the considerable
conceptual difficulties he has to deal with when trying to explain clearly his
novative ideas.
