Equilibrium of balances with unequal arms and weights
The pole as centre of natural and accidental gravity
Leonardo continues his study of the balance, drawn here with beams, semicircular graduated arcs, plumb-lines and hanging weights. He argues that whatever the height at which the arms sit, only the proportion of the arms and their distance from the central line matters, and that arms which resist each other's descent keep that resistance when equal weights are added at equal distances from the pole. Further propositions locate the centre of accidental gravity at the fourth of the pyramid's length toward the base, and show that in a balance the pole holds both the natural and the accidental centre of gravity of the beam and its weights.
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Only the proportion of the arms matters, whatever their height
Even when the arms of the balance sit at different heights, nothing is to be considered but the proportions of the arms, that is how far their ends move away from the central line.
Equal powers with equal weights keep equal resistance
Arms that resist each other's descent, however varied in length, thickness, weight and shape, resist equally at parts equally distant from the pole. If equal powers are given equal weights, their resistances remain equal among themselves.
Centre of accidental gravity at the fourth of the pyramid
For balance a b, let n be the centre of its accidental gravity, at the fourth of the length of the round pyramid toward the base. Hanging equal weights a and m at equal distances from the pole n does not remove the prior equality of the balance.
The pole holds the centres of natural and accidental gravity
In a balance of equal arms with equal weights, the pole contains the centre of the natural gravity of both beam and weights. In one of equal arms but unequal weights unequally placed, the pole holds the natural gravity of the beam and the accidental gravity of the weights.
Graduated balance beams and plumb-lines
The leaves are covered with balances shown as beams pivoting at a pole, several with a semicircular graduated arc and a hanging plumb-line marking the vertical, and with small weights suspended at the arm ends.