This chapter offers a survey of the ways in which the British Board of Longitude handled the range of schemes and projects that were presented by mathematicians and mariners, inventors and entrepreneurs during its final decades to 1828. Labels of impracticality, eccentricity and derangement have long been assigned to many of these proposals, notably in the classification scheme imposed by Astronomer Royal George Airy in his reorganisation of the Board’s archives from the 1840s. This chapter favours close reading of the ways in which schemes were assessed and managed at the time. In the bulky correspondence received, schemes for new devices, calculation methods or navigation techniques were mixed with projects for squaring the circle or endless mechanical power. The Board distinguished between those projects reckoned impossible or unsound, and those it judged irrelevant or beyond its scope. It is shown how much discretionary power the Board exercised, and how its accumulated papers preserve a wide range of protagonists’ technical and scientific interests.

The work–energy theorem in a general reference frame is presented. It relates the change of the mechanical energy of the system in a finite time interval to the impulse work of the interaction forces (and inertia forces, if the frame is a non-Galilean one). Different expressions for the calculation of the kinetic energy of a rigid body are provided. The concept of conservative system and conservative force are introduced, and from them, that of potential energy. The particular case of gravitational energy and energy associated with linear springs is studied. A section is devoted to the study of equilibrium configurations, and their stability, of systems of one degree of freedom. Finally, the impossibility of perpetual motion is proved. The rotation stability of a free rigid body is analyzed through Poinsot’s ellipsoid in an appendix.