Balancing a composite rod against a simple rod
When shorter arms hang nearer the pivot the beam stays level
A horizontal beam at the top of the leaf carries a composite rod and a simple rod hung so their balance can be compared. The proposition states that the midpoints of the weight and length of the two rods stay in equal counterpoise when the middle of one is as much nearer the pivot as it is shorter than the middle of the other. Leonardo adds a proportional rule: multiplying the arms of the simple rod by itself yields its opposite composite rod.
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Equal counterpoise of unequal hanging rods
The middle of the weight and length of the simple and composite rods stand in equal counterpoise on the balance when the middle of one is as much closer to the pole (pivot) b as it is shorter than the middle of the other. With composite rod d e and simple rod e o, the mid-point n of the composite lies nearer b than the mid-point m of the simple by the same amount that the half composite rod is shorter.
Squaring the arm to find the opposite rod
Leonardo closes with a proportional recipe: multiply the arms of the simple rod by themselves, and the result gives its opposite composite rod. The rule ties the lengths of the two balanced rods together arithmetically.