Centre of gravity of the pyramid; levers, beams and loads
The true centre near 7/24 of the length, with problems of levers, dragging and balance
On this leaf Leonardo refines the centre of gravity of a wedge-like "pyramid": its accidental centre lies at one third of the length, but the true natural centre falls between the third and the quarter, about 7/24, or nearer 224/768, a value he calls indemonstrable because it divides into infinite parts. The surrounding diagrams work through mechanical loading problems: a rod swung to successive positions on a quadrant fitted with a toothed wheel, a weighted arm and what its support can sustain, a horse asked to move a load at m or at n, and a beam representing a man dragging a weight, harder at b than at a by a measured 1/8.
On this page
True centre of gravity of the pyramid at about 7/24
The middle of the accidental weight of any pyramid is at one third of its length, but the true centre of the natural weight lies between the third and the fourth, that is 7/24, or nearer 224/768. Leonardo adds that the proper truth is indemonstrable, since it runs off into infinite partitions.
Rod swung through positions on a toothed-wheel quadrant
Leonardo moves rod b through the sites c, d, e, f to learn what its support h p feels in each. He advises beginning at rod f for the sake of the toothed wheel, which goes more easily up than down, and tracks where the perpendicular of the weight loads the base, so that b loads at a, c at n, d at m, e at o, and f at p.
Weighted arm and what its support sustains at K
Placing a weight at the extremity of the rod when at K, the support part m feels the weight at K plus as much again for its own counterpoise n m. If the support weighs less than f, f falls; if more, m sustains twice the weight K and no more, save twice the weight of the rod.
A horse moving a weight at m or at n
Beside a figure lettered a, 2 m, 2 n, Leonardo asks: if a is a horse that moves a weight, which shall we say it moves more easily, the weight placed at m or the weight at n?
Spiritual and real lines proportioned to the weights
The proportion that the spiritual line m d has with the real line m c is the same that the weight of c has with the weight of S, by the 28th conclusion of the 2nd book; and the spiritual line S o does the like with the real line S t.
A man dragging a weight: harder at b than at a by 1/8
The beam figuring a man who drags a weight is defined opposite; it remains to define the dragged thing, which gives more labour to its mover at b than at a, by as much as the part of the line r t enters into the whole c t, which is 1/8.