Centers of gravity of wedges and pyramids
The three centers of a solid, doubling the cube, and locating a pyramid's balance point
Leonardo works out the centers of gravity of solid bodies, distinguishing three centers, of magnitude, of accidental gravity, and the elusive center of natural gravity that halves a body by weight and quantity. He fixes the natural-gravity center of a triangular pyramid, notes the ratio for doubling the cube, and then, for a wedge (conio) whose enclosed pyramid is one third of it, derives the pyramid's center by a converse proportion of space to weight. The right leaf carries the lettered wedge, pyramid, and box diagrams that support the construction.
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The three centers of a solid body
Leonardo distinguishes three centers within 'uniformly non-uniform' bodies: the center of magnitude (midway in length, breadth and thickness), the center of accidental gravity (amid the parts that balance one another), and the least-known center of natural gravity, which divides the body into two parts equal in weight and quantity. The centers of accidental gravity alone cannot locate the natural one.
Center of natural gravity of a triangular pyramid
The pyramid on a triangular base has its center of natural gravity on the line from the middle of the base to the middle of the opposite side, set equidistant from where base and side meet. A solid with equal opposite parts has its natural-gravity center coincident with its center of magnitude.
Doubling the cube
A stated proportion for the Delian problem: the line b c cubed makes a cube double the cube of the line f g cubed. It anchors the ratios used for the solids drawn alongside.
Locating a pyramid's center of gravity from the wedge
For the wedge (conio) a b c the centers of magnitude, accidental gravity and natural gravity fall on lines d e, n o and f g. Knowing the wedge's center and that the enclosed pyramid c d e is one third of the wedge c d e f, Leonardo sets a converse proportion of space to weight, halves the space a g, and places the pyramid's center of natural gravity at i k.
Method stated: find the pyramid's natural-gravity center
He states the aim explicitly: from the known centers of natural gravity of the wedge and of the remainder of the largest inscribed pyramid, to determine the center of natural gravity of that pyramid. The three worked figures carry out the construction.
