Finding the centers of gravity of a cone and its pyramids
Rules relating natural and accidental centers of gravity, with a quarter-circle geometry proof
This opening sets out a series of rules for deriving the natural and accidental centers of gravity of a solid from those of the two parts into which it can be resolved. On folio 111v the rules are applied to a wedge (conio) and its pyramids, with the points c, b and d marking their natural centers of gravity. On folio 108r a large quarter-circle inscribed in a squared grid supports a geometric argument — that the square on the circle's tangents is double the square touched at its inscribed angles, and that the arc n m o is a quarter of the circumference — which Leonardo ties to the centers of magnitude (s t), of accidental gravity (o p) and of natural gravity (d e) of a conical body a b c.
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Rules linking natural and accidental centers of gravity
Five reciprocal rules: from the accidental centers of gravity of the two parts of a body one finds the natural center of the whole; from the natural centers one finds the accidental center; and conversely, from the accidental center of the whole one finds the accidental centers of its two parts.
Centers of gravity of a wedge and its pyramids
c is the natural center of gravity of the wedge a c r m; b is the natural center of gravity of the truncated wedge a n c c; and d is the natural center of gravity of the pyramid c n r.
Quarter-circle in a squared grid and the doubled square
The square whose sides touch the circle is double the square the circle touches at its angles; the triangle a b c is halved by the line d e; the right angle at a makes the curve n m o a quarter circumference, and the side r t is double a r, so the right triangle a r t is double the right triangle a n o.
Centers of magnitude and gravity of the conical body a b c
a b c is the conical body; s t is the center of its magnitude, o p the center of its accidental gravity, and d e the center of its natural gravity; K h and g i are the natural and accidental centers of the largest pyramid the cone can contain.