The Book of Mechanics

...when, however, this line is perpendicular to the horizon, then it is exactly the same (as we just said) as if the weight were sustained exactly at the center of gravity. Wherefore, if the gravity of a weight can hardly be perceived except at its center of gravity, the weight is certainly properly gravitated at that point.

The center of figure among mathematicians is the point from which the semidiameters issue, or through which the diameters pass; such as the center of a circle, and of an ellipse, and also of opposite sections.

The center of magnitude, however, is that which occupies the middle of a figure, or which is equally distant from the outer surface, such as the center of a sphere.

Finally, the center of the world is the point situated in the middle of the universe, and the lowest point of all things.

Moreover, for a better knowledge of these, it is to be observed that these centers sometimes all coincide with one another, sometimes only some; and sometimes not at all. But they all coincide, such as the center of the universe, the center of magnitude of the earth (namely, the sphere composed of water and earth, which we shall call "the earth" for the sake of brevity), the center of figure of the earth, and the center of gravity of the earth. Since the earth is spherical (as all confess), its middle will be the center of figure, from which the semidiameters issue: and the same will be the center of magnitude, since it occupies the middle of its figure. Furthermore, the same point is the center of gravity of the earth, and since the earth rests in the middle of the world, this center of gravity will be located at the center of the universe, and only in this way can all the centers coincide into one. Although the sphere, which contains the earth and water, is composed of bodies of a different species and different gravity, namely of earth and water; yet it is not the case that its middle does not coincide with the center of gravity. For according to the opinion of Aristotle, the earth consists on all sides around the center of the world; and Archimedes affirms that even the moist remains spherical, whose center is the center of the universe. If, therefore, the earth and water remain, both around the center of the universe, then the center of the world is their center of gravity at the same time, and thus the four aforesaid centers coincide into one single point. That these centers together come into one can be sufficiently clear to anyone...