The authors consider the one-dimensional Neumann problem \[ (N)\quad u''+g(u)=s+h(t),\quad u'(0)=0=u'(\pi), \] where \(g\in C^ 1(R)\) and g' never vanishes on an interval. Suppose that \(\lim_{u\to - \infty}g'(u)=a,\lim_{u\to +\infty}g'(u)=b,\) where \(b\in ((n-1)^ 2,n^ 2)\) for some integer \(n\geq 1\) and \(h\in C^ 1[0,2\Pi]\). The main result is the following: Theorem: (a) if \(aS_ 0\), (N) has precisely 2n solutions, and (b) if \(a\in ((k-1)^ 2,k^ 2),\) for some integer k, \(00\) and \(SS^+\) is exactly \(2(n-k+1)\). This extends an earlier work of the same authors for a corresponding Dirichlet problem.