This paper provides two novel contributions. The former regards the observability of the visual-inertial structure from motion. It is proven that, the information contained in the data provided by a monocular camera which observes a single point-feature and by an Inertial Measurement Unit (IMU) allows estimating the absolute scale, the speed in the local frame, the absolute roll and pitch angles, the biases which affect the accelerometer's and the gyroscope's measurements, the magnitude of the gravitational acceleration and the extrinsic camera-IMU calibration. The latter contribution is the derivation of a new closed form solution to determine some of the previous observable quantities by only using few camera measurements collected during a short time interval and the data provided by the IMU during the same time interval. This closed-solution allows us to investigate the intrinsic properties of the visual-inertial structure from motion and in particular to identify the conditions under which the problem has a finite number of solutions. Specifically, it is shown that the problem can have a unique solution, two distinct solutions and infinite solutions depending on the trajectory, on the number of point-features and on their layout and on the number of camera images. The proposed closed solution is finally used in conjunction with a filter based approach in order to show its benefit.