The science of the balance: weights, arms and the centre of the world
Statics of equal and unequal weights, equilibrium about the pole, and a critique of the philosophers
Both faces are a sustained study of the balance (bilancia), illustrated by many beam-and-weight diagrams. Leonardo argues that every heavy body tends to the centre of the world, that a longer arm makes an equal weight twice as weak, and that equilibrium is kept when the proportion of the arms matches the proportion of the weights and the heavier hangs on the shorter arm. He defines the imaginary central line running from the centre of the balance to the centre of the world, relates obliquity of the arms to resistance, and links equal proportions of weights and arms to the finding of one from the other. On f. 75r he criticises the philosophers, holding that because the pendants of the weights converge toward the centre of the world, the beam of unequal weights never truly rests in equality and the pendants are never joined to the beam at right angles.
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A longer arm halves the effective weight
Every heavy body strives toward the centre, and the more oblique opposition resists more. Where m equals n in weight but has an arm twice as long, m is twice as weak, so m being 1 does the office of 2 against n; adding an equal weight at n restores equal opposed powers about the pole.
The central line of the pole and self-restoring equilibrium
Equal weights equally distant from the central line of the pole do not move the arms from equality, and if displaced they return of themselves. The central line is an imaginary straight line from the centre of the balance to the centre of the world, dividing the weights with equal power.
Heavier weight on the shorter arm balances
If the arms of the balance are in the same proportion as the attached weights, and the heavier hangs on the shorter arm, the weights are equally heavy according to their position and the balance rests in equality.
Finding arms from weights and weights from arms
From the proportion of the weights the arms of the balance are found; given the proportions of the arms, the weight is given. If the weight and the part to which it is attached are given, or the weight and the opposite part, then everything is given.
Height measured from the centre of the world
That thing is higher which is more distant from the centre of the world. The motion of the arrow is not of equal height from the centre of the world, and all things of equal height are not situated along a straight line.
Critique of the philosophers on unequal weights
The potential arms of the balance are not in the same proportion as the weights, nor as the real arms, nor are the angles between the pendants and the beam equal. Because the pendants converge toward the centre of the world, unequal weights never rest in equality and the pendants are never joined to the beam at right angles, so the philosophers concluded badly.