This monograph intends to give a general survey of the different branches of the geometry of linear displacements which so far have received attention', The material on this new type of differential geometry has grown so rapidly in re cent years that it is impossible, not only to be complete, but even to do justice to the work of the different authors, so that a selection had to be made, We hope, however, that enough territory is covered to enable the reader to understand the present state of the theory in the essential points, The author wishes to thank several mathematicians who have helped hirn with remarks and suggestions; especially Dr. J. A. SCHOUTEN of Delft and Dr. N. HANSEN BALL of Princeton. Cambridge, Mass., October 1933. D. J. STRUIK. Contents. Page Introduction .... . I. Algebra ..... . 5 1. Vectors and tensors in E n 5 2. Densities . . . . 6 3. Measuring vectors . 7 4. Point algebra. . . 8 5. The general manifold X" 9 6. Non-holonomic measuring vectors . 10 7. Pseudotensors ...... . 12 11. Affine connections .... . 13 1. The principle of displacement 13 2. Affine displacement Ln 14 3. Torsion. . . . . . 17 4. WEYL connection . 18 5. Metrical connection 19 6. Curvature. . . 19 7. Integrability 20 8. Some identities 21 9. Non-holonomic systems 22 10. Transformation groups 23 IH. Connections associated with differential equations 24 1. Paths ........ . 24 2. Projective transformations 25 3. THoMAs parameters . . .