Two approaches (of È. Cartan and of T. Y. Thomas and J. H. C. Whitehead) to the study of projective connections are compared. Spray geometry is reviewed, the Thomas-Whitehead construction and its generalization to sprays are discussed, and afterwards the Cartan theory in the affine case is described. A certain bundle is introduced which realizes explicitly Cartan's idea of attaching a projective space to each point of a manifold, called the Cartan bundle. This bundle is defined independent of any particular choice of connection, but to any projective class of sprays one can associate a unique Cartan connection on the Cartan bundle, as it is shown. In the course of the discussion a Cartan normal projective connection is derived for a system of second-order ordinary differential equations (extending the results of Cartan from a single equation to many) and the concept of a normal Thomas-Whitehead connection is generalized from affine to general sprays. Finally it is described in detail how to derive the Cartan connection from the generalized Thomas-Whitehead data for any projective equivalence class of sprays.