Composite and simple rods in equilibrium
Length, thickness and the square-root rule for balancing rods
Diagrams of composite and simple rods, one part-composite, illustrate how differently built beams can be brought into balance. Leonardo states that the proportion the length of the simple rod bears to that of the composite rod is matched by the proportion of the composite rod's thickness to the simple rod's, so that the two stand in equilibrium. He adds that the simple rod balancing a composite one is always its square root, and that where the arms of a balance are unequal in length at the point of equality, the shorter arm must be correspondingly thicker.
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Balancing a composite rod against a simple one
The proportion the length of the simple rod has to the length of the composite rod is the same as the proportion the composite rod's thickness has to the simple rod's thickness. Under this reciprocal relation the two rods stand in equilibrium.
The simple rod is the square root of the composite
Leonardo states a numerical rule for the balancing rods: the simple rod that stands in equilibrium with a composite one will always be the square root of that composite rod. Length and thickness are thus tied by a proportional, root relationship.
Unequal arms must differ in thickness
A suspended rod supports the composite rod as an experiment, exactly as promised earlier. Where the arms of the balance are unequal in length at the position of equality, necessity requires that an arm be as much thicker as it is shorter.