The Book of Mechanics

...a sphere, for instance of wood or of another (though similar) nature; for its middle will be the center of magnitude and the center of figure, and the same point will be its 16 Federicus Commandinus on the center of gravity of solids. center of gravity, around which the parts are in equilibrium on all sides. And since this sphere is not at the center of the world, therefore only three centers will coincide. But if the sphere were not similar, but dissimilar, for instance with one half being lead and the other half wood, then its middle would indeed be the center of magnitude and figure, but by no means the center of gravity. For the parts could not be in equilibrium around the middle on all sides; rather, the center of gravity would decline toward the heavier part, namely the leaden one. And in this way, only two centers will coincide. This also happens (though in a different manner) in an ellipse, whose center is the center of figure, since the diameters pass through it; and the same point 4. Fed. Command. on the center of gravity of solids. is its center of gravity. Because this is not strictly the middle of the figure, it will not be the center of magnitude either. For the middle of the figure pertains strictly only to a circle and a sphere.

in the second book of this Wherefore, two centers coincide simultaneously in this way as well. In the figure of a parabola bounded by a straight line, the center of gravity is found within the figure, as it can be neither the center of figure nor the center of magnitude. Indeed, in this figure there can be no middle, whence no center of magnitude will exist, and since the diameters in a parabola are equidistant from one another, as is evident from the first book of the Conics of Apollonius of Perga, no center of figure will exist either. Thus, therefore, the centers will not coincide in any way.

It is also necessary to know that the center of gravity is more common and found in more things than the centers of magnitude and figure; and the center of figure is more common than the center of magnitude. For any body and any figure must have a center of gravity, either internally or externally. Internally, as the center of gravity of some regular body, which is in the middle of the figure, or of some figure like A, whose center of gravity is on the perimeter of the figure, as at point B; externally, however, as figure C, whose center of gravity is external, as at D; which is to be understood if the heavy body C were to tend toward the center of the world.