  The Newtonian limit of the most general fourth order gravity is performed
with metric approach in the Jordan frame with no gauge condition. The most
general theory with fourth order differential equations is obtained by
generalizing the $f(R)$ term in the action with a generic function containing
other two curvature invariants: \emph{Ricci square}
($R_{\alpha\beta}R^{\alpha\beta}$) and \emph{Riemann square}
($R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}$). The spherically
symmetric solutions of metric tensor yet present Yukawa-like spatial behavior,
but now one has two characteristic lengths. At Newtonian order any function of
curvature invariants gives us the same outcome like the so-called
\emph{Quadratic Lagrangian} of Gravity. From Gauss - Bonnet invariant one have
the complete interpretation of solutions and the absence of a possible third
characteristic length linked to Riemann square contribution. From analysis of
metric potentials, generated by point-like source, one has a constraint
condition on the derivatives of $f$ with respect to scalar invariants.
