The balance: arms, poles, and centres of accidental gravity
Statics of equal and unequal balance arms — equal weights at equal distances from the pole resist equally.
Both leaves pursue the theory of the balance (bilancia). Leonardo argues that whatever the heights, lengths, thicknesses or shapes of the arms, only the proportion of the arms and their distance from the central line count, and that equal weights hung at equal distances from the pole resist equally. A worked proof takes a balance a b whose accidental centre of gravity n lies at the fourth of the length of the round beam toward the base, showing that equal weights hung at equal distances leave its equilibrium unchanged. The drawings show balances with suspended weights and a fan-shaped graduation of angles.
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Only the proportion of the arms matters
Even though the arms of the balance stand at different heights, nothing is to be considered of them but the proportions of the arms and how far their extremities move from the central line. Height above the ground does not alter the reckoning of the arms.
Equal weights at equal distances resist equally
Arms that of themselves resist one another's descent, however various in length, thickness, weight and shape, do not fail — in the parts equally distant from the pole — to resist equally the equal weights attached there. Equal powers given equal weights keep their resistances equal.
Proof: hanging equal weights leaves equilibrium unchanged
Let the balance be a b, with n the centre of its accidental gravity at the fourth of the length of the round beam toward the base. Hanging equal weights at equal distances a and m from the pole n does not remove the first equality, so the arms remain in their former degree of mutual resistance.
Where the centres of gravity gather at the pole
In a balance of equal arms with equal weights, the pole receives within itself the centre of natural gravity both of the balance and of the weights. With equal arms but unequal, unequally placed weights, the pole holds the natural centre of the balance and also the accidental centre of the weights.
Accidental centre of gravity at the pole n1
A balance of unequal arms and unequal weights has the centre of the accidental gravity of itself and its weights at its pole: that is, n1 is the centre of the said accidental gravity.
Balances and a graduated fan of angles
The sheet is filled with drawn balance-beams carrying weights on cords, together with a semicircular fan of radiating lines used to measure the angles the arms make. These instruments illustrate the statics worked out in the text.