These are sections 1 and 2 of UFT370 , entitled “Cartan Geometry and the Gyroscope”. The paper defines the Cartan spin connection of classical rotational dynamics in general, and several examples are worked out for the nutations and precessions of the freely rotating asymmetric top, and for the asymmetric top subjected to an external torque. Full details are given in extensive calculations in the accompanying notes, calculations which have all been checked by computer algebra as usual. Both the Euler angles and spherical polar coordinates are used. Section 3 is pencilled in for computations and graphics by co author Dr. Horst Eckardt, and for his dumbbell model of the nutations and precessions of the earth in orbit around the sun. This may lead to a more complete theory of the Milankowitch cycles. With the theory of this paper, the Laithwaite and Shipov experiments can be fully understood and simulated by a desktop computer.