In this note we take up anew the old problem of the existence of closed geodesics on an n-dimensional spherical manifold M, i.e., a riemannian manifold which as underlying manifold has the n-sphere S". Our goal is to construct closed geodesics on M with the help of the sPace of circles on S". This has been done before. After Lyusternik [6] had proved the existence of n closed geodesics in this manner, Alber [1] claimed to have proved the existence of at least g(n) closed geodesics obtainable from the space of circles. Here, g ( n ) = 2 n s 1 with s determined by O < s = n 2 k < 2 k. As was pointed out recently by W. Ballmann, Alber's proof is based on an incorrect statement on the cohomology of the space of non-constant unparameterized closed curves, cf. also [4] where we repeated Alber's mistake. In this paper we will show how the situation can be remedied by working with the cohomology of the space of circles instead with the cohomology of the bigger space of all closed curves on the sphere. We will use extensively the Hilbert manifold A M of closed Hi-curves on M, together with its riemannian metric and the gradient flow stemming from the energy integral E" A M ~ IR. To keep this note short we refer for this theory to our monography [41 or our forthcoming book [5]. Here we only recall the definitions.