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Introduction Part I. The Unconnected Manifold: 1. Invariance 2. Integrals Part II. Affinely Connected Manifold: 3. Invariant derivatives 4. Some relations between ordinary and invariant derivatives 5. The notion of parallel transfer 6. The curvature tensor 7. The geodesics of an affine connexion 8. The general geometrical hypothesis about gravitation Part III. Metrically Connected Manifold: 9. Metrical affinities 10. The meaning of the metric according to the special theory of relativity 11. Conservation laws and variational principles 12. Generalizations of Einstein's theory.