It does not seem to be sufficiently realised that most of the theory usually classed under “Centre of Gravity” is purely geometrical in cliaracter ; and, on the other hand, such impossible terms as “centre of gravity of an area” still offend the eye. This paper suggests a presentation of the subject free from such defects ; but its main object is to establish a sounder transition from the treatment of discrete point-systems, which is the essential groundwork of the theory, to the much further reaching treatment of continuous point-systems. The fact that this transition remains in an unsatisfactory state is an instance of the reluctance to face the necessity for explicit recognition of certain fundamental ideas of the Infinitesimal Calculus, in elementary Mathematics. It is not long since the proper conception of tangent was admitted as suitable for an elementary course of Geometry. Still more recently the more abstruse conception of arc-length has been faced. From what follows it may appear that the conception of centroid awaits its turn.