Summary: Observability of a general nonlinear system -- given in terms of an ODE \(\dot {x}=f(x)\) and an output map \(y=c(x)\) -- is defined as in linear system theory (i.e.\,if \(f(x)=Ax\) and \(c(x)=Cx\)). In contrast to standard treatment of the subject we present a criterion for observability which is not a generalization of a known linear test. It is obtained by evaluation of ``approximate first integrals''. This concept is borrowed from nonlinear control theory where it appears under the label ``Dissipation Inequality'' and serves as a link with Hamilton-Jacobi theory.