The Center of Gravity of Uneven Bodies
Why the center of a body's shape need not coincide with its center of natural gravity
Under 'On gravity,' Leonardo argues that if a heavy body does not have uniform opposite sides around the center of its magnitude, that geometric center will never coincide with the center of its natural gravity. He proves the point with a rectangle whose opposite sides are equal: there the center of magnitude n is also the center of gravity. He then contrasts a triangular (pyramidal) figure, where the apex angle a is unlike its opposite side b c, so the center of length and breadth S falls away from the true center of gravity, which lies on the line f g.
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Uneven sides split the two centers
If the heavy body does not have uniform opposite sides around the center of its magnitude, the center of that magnitude will never be concentric with the center of its natural gravity.
Proof on the balanced rectangle
Let the heavy body be a c d f, whose opposite sides a c and d f are uniform and whose other two opposite sides a d and c f are uniform; it follows that the center of its magnitude, n, is also the center of its natural gravity.
The pyramidal figure with unequal angle
But in the first, pyramidal figure, if the angle a is not similar to its opposite side b c, even though the sides a b and a c are similar, the center S of length and breadth is not the center of natural gravity, which lies in the middle of the line f g.