We discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential equation
\begin{equation*}
-\Big( u'/{ \sqrt{1-{u'}^2}}\Big)'
= f(t,u).
\end{equation*}
Depending on the behaviour of $f=f(t,s)$ near $s=0$, we prove the existence of either
one, or two, or three, or infinitely many positive solutions. In general,
the positivity of $f$ is not required.
All results are obtained by reduction to an equivalent non-singular problem to which variational or topological methods apply in a classical fashion.
